April 04, 2023
Pam Harris
Episode 146

Math is Figure-Out-Able with Pam Harris

Ep 146: (Mis)Understanding the Area Model

Info
Share

Math is Figure-Out-Able with Pam Harris

Ep 146: (Mis)Understanding the Area Model

Apr 04, 2023
Episode 146

Pam Harris

The area model is a great tool to help students develop multiplicative and spatial reasoning, but let's get clear on some things. In this episode Pam and Kim discuss some of the common misconceptions about the area model and show how it can be used help students make connections between area and multiplication.

Talking points:

- What is an area model?
- Common misunderstanding around an area model
- Place Value Partial Products vs Smart Partial Products
- Rote memorizing perspective vs Math-is-figureoutable perespective
- Closed/Discrete versus Open arrays
- Area models build spatial reasoning and connections of products to area and factors to dimensions
- Ratio Tables are better models for cranky computation problems

See Episodes 129-133 for more about Ratio Tables and 147 and 148 for more about 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

Listen on

Share Episode

The area model is a great tool to help students develop multiplicative and spatial reasoning, but let's get clear on some things. In this episode Pam and Kim discuss some of the common misconceptions about the area model and show how it can be used help students make connections between area and multiplication.

Talking points:

- What is an area model?
- Common misunderstanding around an area model
- Place Value Partial Products vs Smart Partial Products
- Rote memorizing perspective vs Math-is-figureoutable perespective
- Closed/Discrete versus Open arrays
- Area models build spatial reasoning and connections of products to area and factors to dimensions
- Ratio Tables are better models for cranky computation problems

See Episodes 129-133 for more about Ratio Tables and 147 and 148 for more about 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 y'all it's about making sense of problems. I was going to say "since making".That will work too. Noticing patterns and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.Woah!

Kim:So, Pam, I was checking out podcast data the other day, and I saw that we got this really sweet comment that I hadn't seen before, and I don't know if you have. Can I share it before we get started today?

Pam:Yeah, absolutely. I like sweet comments.

Kim:So, there is a new comment on the podcast that says, "Thank you, Kim and Pam for helping me love doing Real Math and teaching math to my scholars.Doing Real Math is the missing piece. As I begin to play with a relationship with numbers myself I'm becoming a better teacher.Watching my scholars' eyes light up as things pop for them is why I teach. Keep on spreading the message that math is truly figure-out-able! We are listening!"

Pam:Whoa!

Kim:And that's from a listener called OneRedJeepGirl. So,thanks so much for listening.

Pam:Well, OneRedJeepGirl,you're amazing, and we sure appreciate you listening. And thanks for that feedback. That's awesome! What a great way to start the day. Nice. Nice. Yeah,scholars eyes lighting up is definitely the reason we teach,right?

Kim:Yeah.

Pam:And if we can actually then teach Real Math, that is amazing. Super important.Awesome.

Kim:Yeah, really cool.So, today, we're going to kick off a short series about a model that we love and find so much value in. But we think it's gotten a little bit of a bad name as people have tried to make use of it. So, today's topic is all about the area model.

Pam:The area model. Yeah, so that shouldn't be as tricky as it is, but it's a little bit.There's some subtle things happening that we would like to parse out today. So, let's start with what do we mean when we talk about the area model.

Kim:Yeah.Well, and it's actually... You know, some people who haven't done a lot of work with it might be familiar with it because it's the model of choice that people complain about when they're on social media, and they're complaining about "math these days", and they are quoting"math these days", so you've seen it. It's a rectangle, but it could be a square depending on the dimensions. And it's what people sketch when they're solving a multiplication problem. So, if I were asked 18times 25, I would write 18 on one dimension of the rectangle and 25 on the other dimension of the rectangle. So, it would be a little bit longer. And then, I would consider cutting that problem up in different ways to solve the parts of that problem.So, that representation is the area model.

Pam:Yeah, interestingly, you just really said that you're going to cut up that rectangle in intelligent ways, to solve parts of the problem, to then help you kind of solve the whole problem, right?

Kim:Mmhmm.

Pam:So, you've just talked about labeling the length and the width of a rectangle, and by doing so you're actually then...To solve a multiplication problem. You're saying, "If this is length by that width..." then you're really sort of counting all of those unit squares. That if you have an 18 by 25, then you would have these little tiny1 by 1 squares, and you would have 18 by 25 of them. And as you're counting those little unit squares, that's the area of that rectangle, right? So, you have the length and the width of a rectangle, and all those unit squares. Added up together,that's the area. So, the dimensions of the rectangle are like the factors in a multiplication problem. And the area of the rectangle is like the product of that multiplication problem, so we have this correlation between factors and products, and dimensions and area, right?Okay, so, we've though seen people do some rote memorizable stuff with what they're calling an area model, but they're actually using it more as a procedure. So, for example, we see an 18 by 25. Someone's solving the problem 18 times 25,and they've drawn a square. And we are like, "But hang on. The length of 18 should be..." In fact, I just grabbed my pen. I don't know if you can hear me click it there. There, I'm clicking. Because I'm a pen person, Kim, not pencil.

Kim:You can be wrong.

Pam:I drew a length of 18. When I draw the other side of the rectangle as 25, If that's the problem I'm going to represent as the rectangle that's 18 by25, that 25 should be longer than the 18. Now, not like twice as long, right? Twice as long and be 36. It's not 36, it's only 25. But it's also not the same length as 18. It certainly shouldn't be shorter than 18.The typical thing that we see that kind of is going awry is we see people drawing a square, not a rectangle. And in this case,it should be a rectangle because18 and 25 are not the same length. So, if I see someone solving a multiplication problem using a square when the dimensions aren't square, then instantly I have a little bit of a... What do I want to say? The hair raises on the back of my neck. It's a little bit like,"Mmm..." I start to twitch just a little bit. I'm like, "Umm,so..." It appears to me, when people start that way, that then they're heading down a road of rote memorizable or what I might call fake mathing. It's not really fake math, but it's definitely fake mathing. I'm not doing the mathematizing that a mathematician would do. I might be using that square as more of a graphic organizer to keep track of steps, to keep track of a procedure.

Kim:Right.

Pam:So, I'm going to sort of talk about what it might look like because we're just... We're not visual here. We're just audio. So, that 18 by 25.Somebody might draw a square,and so I just drew a square, and I put 18 on one side, and I put25 on one side. And then, they tend to cut that square evenly into 4 chunks. Again, if I'm cutting... Well, I'll let me keep going. If I cut it into 4chunks, then we often see that they label the first sort of cut on the 18 side. They'll label that 10. And then, the second cut, they'll label 8. Because they cut the 18 into place value parts. 10 and 8, 18. 10 and 8.And then, for the 25, they'll cut the 25 into place value parts, 20 and 5. But there's a problem. If I've just cut that square. Remember, it shouldn't have been a square, should have been a rectangle. 25 is longer than 18. But they've drawn a square, and then they've cut the18 into 10, and then cut the 25into 20. Now, I have this other square because I've cut that big square into 4 squares. So, I have that top left square now labeled as a 10 by 20. Well, it shouldn't be a square if it's a10 by 20, right? Like it should be... If the one side is 10,then the other side should be exactly twice as long because that's 20. So, we now have lots of things that are mislabeled.We're calling it an area model or an area diagram, but we're not really representing area at all.

Kim:Right.

Pam:What we're doing is we're,if we're doing it this way, we are chunking the multiplication problem into place value chunks,and then we're kind of saying"Alright, so now I'm looking at this particular chunk, and I'm thinking that that's a 10 by20." But it's not really a 10 by20 because I haven't represented that way. But I'm thinking 10times 20. So, then, I think about 10 times 20, and I might fill that in with 200. And then,I'm looking at the one next to it. That will be a 10 times 5.It should be a 10 by 5, which should be now twice as high as it is wide right? 10 by 5. But I've represented it as a square because I'm not thinking about area model. I'm used to using it as a way to keep track of the steps. So, then that looks like50. And then, I'm looking at the8 by 20. That will be 160. And then, I'm looking at 8 by 5, and that would be 40. So, basically,now I have a square that's split into 4 squares. And then, in each of those squares, I have a partial product. And that's the definition of what we mean by partial products. We've done part of the product. We're looking for the total product of18 by 25, but we have these partial products in each of these squares. And then, the kids add those up together. So,what you've done, listeners, if that's what you've been using and calling it an area model, if you've been using this square to find the product of factors,then you've kind of been using it as a graphic organizer.You've said to yourself, "I'm going to find these partial products and very specifically place value partial products,and I'm going to put it in this square to kind of keep track, so that I don't miss one of them."I get all 4 of the partial products that I've split by place value, and then you add those up together, and you get a product. If we could just maybe agree, it's not an area model.Could we agree that if that's what you've done, you've used a graphic organizer to make sure that you've gotten all of the numbers split into place value,and then you multiply those together, and you've added them up, but what it's not is an area model. It's not representing area because you drew a square.And then, it's not a smart area model if when you cut it up, you always cut it in by place value.If the way you split up the dimensions for 18 is always 10and 8, and the way you split it the 25 is always 20 and 5, then that's not a very clever way of finding partial products.Especially for a problem like 18times 25. What could be a better way to do that? Well, Kim, I'm just going to ask you. I'm going to ask you. If I gave you 18times 25, what partial products might you be inclined to use?Would you just right now do 10and 8, and 20 and 5, and then find those 4?

Kim:No, no, no.

Pam:What would you do?

Kim:I think off the top of my head right now, I think do just split one of the numbers. So, I think I would just split the 18,and I would say 10 times 25. And then, 8 times 25. So, my area model would just have one cut.

Pam:And I just drew a 10 by something that's about 2 and a1/2 times wider. 10 by 25. And then, I tacked on the bottom of that an 8, which is shorter than the 10 by the same 25. So, I now have one rectangle that's an 18by 25 that actually looks like an 18 by 25. And like you said,it only has one cut. I only cut the 18 into 10 and 8, but Kim.

Kim:Yeah?You might know 10 times 25. I mean, that's. Okay. 10 times 10.We can do that. That's... Go ahead. What is 10 times 25?Two hundred fifty.

Pam:That's 250. But you don't know 8 times 25? That's silly.Like, you wouldn't know that.

Kim:Well,so I knew 8 quarters. But even if you didn't know 8 quarters,if you know ten 25's, then figuring eight 25's it's not so bad.

Pam:It's just 2 less.

Kim:Two less 25's. Yeah.

Pam:(unclear). Oh, and then 250minus 50, that's just 200. Cool.Hey, tell me about 8 quarters because that we have listeners out there who don't think in terms of quarters. So, do(unclear).

Kim:Yeah, so4 quarters is $1, so then 8quarters would be $2, which is200 cents.

Pam:Nice. So, 8 times 25 would be 200. And then, you could add that 250 and that 200 together to get 450. And you're saying that's 18 times 25. And that would be a a smart partial product, not just a place value partial product, but like a clever partial product where you've really thought about those relationships. So, when you think that math is rote memorizable, then you might see someone saying, "Use an area model." And you might even ignore the name. You might just like, "Oh, what do you call that? You call it an area model.I don't know why, but okay.That's its name. It could be the George model. It could be the..." Help me. Give me another name. "The Maria model." Like,it could be whatever, right? It could just be a name that we've given. "Oh, okay." So, if I come at this model from a...Especially if I come at it from a CRA perspective. But if I come at it from the perspective of,"Oh, this is another new thing you want me to teach? I don't know why because math is rote memorizable, and I know kids just need the steps. All right,fine. You're telling me do this other thing. Okay, what? Let's see, it looks like what you're doing is you're drawing a square, and you're cutting things by place value. Oh, well,that's kind of interesting."Then you're making me actually think about the place values.Like, that 10 times 20. I had to think about 10 times 20 a little bit. I have to know something.It's not just the digits that we're used to in the traditional algorithm. In the traditional multiplication algorithm, it's always just digits multiplying and that magic zero kind of takes care of the place value as you shift things over on the second line.

Kim:Mmhmm.

Pam:And so, the algorithms kind of taking care of that place value. So, you might, if you come at it from a math is rote memorizable perspective, you might look at this model that has a name. I don't know. The area. I don't know if that means anything. And you might be like,"Oh, well, this is a way for the kids to actually have to think about the place values involved.Huh. Okay. Alright. And then they add them up. I mean, that seems kind of less efficient than the traditional algorithm,but if you want me to teach it,I guess I can. 'Alright,everybody, here's how you do it.'" And then you turn it into steps because that's what math is. If you're coming from that rote memorizing perspective.Again, no blame here, right? I'm not blaming. Most of us were taught that that was the perspective, so I can completely understand that if that's your perspective, you would come upon this model, the George model,the Maria model, and you would say, "Oh, okay. Well, now we're going to like do this every time. Okay. 'Students, here's what you do. Draw a square, cut the numbers by place value on each side. Now, do the multiplications in each of these bits. Now, add them together.'Woah! Okay, we've done that thing. Don't really know why we did that thing, except they kept the place values alive. Okay,fine." And can you see maybe then you made the connection between those chunks and how it looked on the traditional algorithm. And you might be like, "Okay, I don't know why we had to do that, but alright,we're done," if you're coming at it from a rote memorizer perspective. But if you're coming at it from a Math is Figure-Out-Able perspective,what real mathematicians do,what people do when they're mathing, then you might be more like Kim. "Oh, well, if I can think about an 18 by 25, then I can cut it in a way that I can then find those products in a more efficient, clever kind of way. I can use that model to help me think and reason about how I'm using the nice relationships to solve that problem."

Kim:Yeah. So, I'm thinking about when I was doing some coaching in my own school, which is always interesting. You know,I had done a lot of work in my classroom with area models, and I just assumed that other people thought about them in the same way that I did, which was a poor assumption, right? Very poor assumption. And I remember going into a classroom and seeing a teacher do exactly what you're describing, square and cutting up into chunks, and my favorite...

Pam:Cutting it up into the same chunks, right?

Kim:In chunks, same chunks,but...

Pam:Four squares, a square cut into 4 squares. Mmhmm.

Kim:But the best is when you come across a problem where it's like 40 times 20, and kids are dutifully drawing a square,cutting into 4 chunks, and then they write the 4, 0 for the 40on one side and the 2, 0. Like,all the zeros, right? So,(unclear).

Pam:Hey, I got one better for you. I got one better for you. I was at a campus. It was probably about the same time. And I saw a teacher, or I saw a kid really,supposed to multiply something like 5 times 14, drew a square and cut the square to 4 chunks.On the 5 side wrote 0 and then5, and on the 14 wrote 10 and 4and then dutifully multiplied 0times 10, 0 times 4, 5 times 10,5 times 4. Like, dutifully multiplied. I was like, "What?"It's so not like it could be.The point is, listeners...Sorry, we're not trying to be awnry. We're trying to say, Ah,that's a misunderstanding that it's supposed to actually represent area. So, if it was 5by 14, it should be a length of5 by a longer length, like more than twice as long, almost three times as long 14. And then, it's not cut into 4 chunks, right?It's just a 5 by, and then you could maybe do 5 by 10. And 5 by4.

Kim:Yeah.

Pam:Like, those two chunks.Sorry, I interrupted your story.

Kim:That's okay. No, it's fine.You understand what I was saying because you have a similar scenario. So, before we move on,I'm just going to wrap up. You just said a whole lot, and so I'm going to wrap up. If we're talking about a graphic organizer versus area model. So,a graphic organizer is really only helpful if you're going to proceduralise the place value partial products. Like,that's... You might as well call it a graphic organizer at that point. But if you're talking about an (unclear)

Pam:Please don't call it an area model.

Kim:Right.

Pam:Like, that's what it is. If you're like, "Hey, every time you're going to see a problem,do this procedure, "then fine,if that's your goal, but then call the graphic organizer.Sorry, keep going.

Kim:But if you're going to call it an area model, and you're going to think about the area,and you're thinking and reasoning, you're probably not going to do place value chunks,and so there would be different cuts or chunks in your model.And heavens, let's make it proportional.

Pam:Yes, please. Yeah, nice.So, let's get clear on a few terms at the end of this podcast here. When... Nah, let's just go this way. I was going to do a whole history thing, and I won't. Let's get clear on some terms. So, if you have a discrete array of things. If I have a 4 by 5 array of marbles,like where I literally lineup 4rows by 5 columns of marbles,then that, the term for that is a "discrete array". It's an array of objects, and they're discrete because there's a marble, and a marble, and a marble. So, I could have a 4 by5 array of marbles. I could have a 4 by 5 array of post it notes.I can have a 4 by 5 array of...Help me? What else? Peanut M&M's That's on my desk over here. I could have a 4... Like, as soon as it's an object, that we call that discrete because it's not a measurement model. It's not a span of things. And so, I could sort of line those guys up, and that's a discrete array of things. Now, as soon as we take that discrete array, and we sort of flatten it out, and we actually are talking about area.And maybe I should have left the post it notes for that. Like, if I had a 4 by 5 array of square posted notes. Which I actually have a square post it note right in front of me right here on my desk. If I had 4 of those all lined up next to each other by 5of them all lined up next to each other. I filled that in, so I had a 4 by 5 of these square units next to each other, then that we call a "closed array".That's a closed array because we've got all the grid marks.So, if you look at grid paper,and you outline 4 rows by 5columns. You outline a rectangle like that, and you've got all the grid marks in the middle,then that is a closed array.It's an area model with all the gridlines. A discrete array could be that same 4 by 5 on that grid paper but where I've put a marble in each of those squares. And then, I'm talking about the discrete array would be the marbles, not the gridlines underneath it. The gridlines underneath it would actually represent the area of that 4 by 5. Then, if we sort of lifted out of grid paper, and I draw that 4 by 5, so it's still4 by 5...it's not a square,should be a little bit longer than it is deep...but I don't put all the grid marks in there,then that's called an "open array". An open array is where we have well chosen grid lines.So, for example, when Kim did 18by 25, and she split it into 10and 8 by 25. That's an open array because we have well chosen grid lines. We only have the grid lines we need. Now, you probably recognize that as a correlation between the open number line. We can have a closed number line. Well, in fact, let me even back up further than that. I can have a line of 10 marbles. I can line up 10 marbles, and I can count those 10 marbles. But as soon as I sort of flatten those out, and I make it more measurement. I make it the distance between tick marks. Now, I have a number line. And if I have all of the tick marks in between 0 and 10,and I'm counting the span between each of those, the measurement from each tick mark to the next one, but I have all of the tick marks, then that's called a closed number line.When I have well chosen tick marks, say I'm going to add 8and 6. And so, then I just draw a number line, and I put the tick mark where 8 is, and then I do a hop to 10 and get to that friendly 10, and I put a tick mark where the 10 is, and then I add the rest of it on, the extra4. Because I'm doing 10 and 6.So, I did 10 and 2, and then 4more. And I do that extra 4, and I have a tick mark. 10 and 4 is14. Those are well chosen tick marks. That's an open number line. So, just like I can have a bunch of discrete objects to a close number line where I have all the tick marks to an open number line where I have well chosen tick marks, similarly with area I can have a... Or maybe I should say with arrays,I can have a discrete array of objects where I have a number of rows by number of columns. But as soon as we flatten that out,we make it more of a span where I'm actually measuring the span of length by a span of length.If you can see my hands right now every time I say "span" my hands are moving apart from each other. So, I'm measuring that distance from tick mark to tick marker or between the grid line to grid line. As soon as I'm measuring that distance from the grid lines that are measuring the rows and the distance that are on the grid lines that are measuring the columns, now we create that two dimensional measurement of that unit by unit square. That's area. That's where area comes from. So, we go from the discrete array of items, to a closed array where we have all of the grid lines,to an open array where we have well chosen grid lines.

Kim:Yeah.

Pam:Cool. So, super interesting. Kim, the other day,I was working on this with a group of participants, and one of the participants in the middle of a Problem String looked up and goes, "Wait, wait.So, if we do this with students,they're not just learning about multiplication, but they're actually learning about area."

Kim:Oh, that's great! What a good "aha" moment for them!

Pam:Totally great "aha" moment!Because in that moment I was like, "Yeah, and you know what that's why it's called the area model." And eyes lit up around the room, and they were like,"Oh, that's why they call it an area model." And I was like,"Yeah, it's not just a name."It's not just a name of this procedure that we're doing at all. Like, it actually is intended to teach area and multiplication at the same time.And it's one of those places where we can connect spatial reasoning and numeric reasoning at the same time. So, in our experience, Kim, there was a few things that worked and didn't work. I just want to share a couple of those. When we started helping teachers understand,really think and reason about multiplication, and we started using the area model, or maybe I would call it an open array, to help teachers and students reason about multiplication,what didn't really work so well is using the area model or rectangle as a tool for thinking about ugly problems. When the problems were ugly, the dimensions were... Like, one of the ones was way too long and way too short, and then cutting it up. And kids kind of got a little bit sometimes, and teachers too could get a little bit... What's the word I want?Stuck. Or kind of like fussing too much with making the rectangle look exactly right,and not just using it to help them think about the relationships.

Kim:Yeah.

Pam:Especially for division.When we started using an area model or an open array to...Maybe I should have done the caveat this a different way. Let me tell you what worked well.What worked well was to help kids reason about area, to help them understand that the span by that span created this two dimensional square unit, and that there was something about being able to measure the amount of carpet that you need by the amount of the square units that you could kind of lay out on the carpet. Building their spatial reasoning was great using the area model, using this open array. Making sense of which chunks to use, making sense of dimensions, and factors, and area, and product, making sense of all that, that spatial reasoning, that was brilliant.What didn't work so well was then after that saying, "Now use the area model as a tool to reason about these cranky problems." That didn't work as well. What we learned was there was a better tool to reason about the cranky problems, well and really to reason about multiplication, after we built spatial reasoning. So, we're not saying get rid of area model and get rid of the open array. We're not saying get rid of it at all.We're saying, that's the place to begin and to build spatial reasoning. But as the tool that we found to actually compute with, to actually find those products, or if you're dividing to find that dividend... No, not find the dividend, find the quotient. I can get that word right. It was better to use a ratio table.

Kim:Yeah.

Pam:Ratio tables became the better tool with which to compute after, and maybe along with, at the same time as, we're building the sense of area, and the dimensions, and the size,and the magnitude, and the spatial sense with the area model. Did I just say any of that well?

Kim:[Kim laughs] I followed you.

Pam:I feel like I was all over the place with that. So, maybe I'll do just a quick recap. So,teachers, do we want you to use the area model and open array?Absolutely. To build spatial sense, to help make sense of chunking products, and to connect dimensions to factors and area to products. We absolutely want you to do that.But we also want you to introduce the ratio table and help build that model because that is going to become the tool that students will use as a tool for computing.

Kim:Yeah. And we have a whole series about the ratio table, so they can check that out, and we'll put the episodes in the show notes. But don't leave us now because we have more conversation to be had about the area model.

Pam:Ooh, and one other quick thing I want to mention. Sorry,sorry.

Kim:Yeah.

Pam:Is that another another good use of the area model...and maybe we'll talk about this more in the next episode, so I'll just mention it here...is making sense of the distributive property. That's a brilliant. We want to make sense of the distributive property using this area model more and more, and we will do that more in upcoming episodes.

Kim:Yeah. So, also, I don't know if you know this. You know this, Pam. But we have merch,and this month we just released a super cool shirt around area and perimeter that we think that you'll love. You can check that out at mathisfigureoutable.com/merchandise.And...

Pam:Merch! Yay!

Kim:Yeah, we are almost at a 5out of 5 on Apple Podcast. So,would you please rate the podcast. It helps more listeners find the podcast, so we can spread the Math is Figure-Out-Able word and everybody can jump in on the movement.

Pam:Yes, please! Give us a rating on there, so more people can find the podcast. Ya'll,thank you 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!

×

All content © 2023 Math is Figure-Out-Able with Pam Harris.