Math is Figure-Out-Able with Pam Harris

Ep 147: The Area Model to Build Reasoning

April 11, 2023 Pam Harris Episode 147
Ep 147: The Area Model to Build Reasoning
Math is Figure-Out-Able with Pam Harris
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Math is Figure-Out-Able with Pam Harris
Ep 147: The Area Model to Build Reasoning
Apr 11, 2023 Episode 147
Pam Harris

The area model can do more than build Spatial Reasoning. It can help make sense of the major multiplication and division strategies! In this episode Pam and Kim use a Problem String to show how reasoning about area can build more sophisticated and efficient multiplicative relationships.
Talking Points:

  • Multiplication is about grouping groups
  • The area model can help make sense of multiplicative relationships and properties
  • Choose and celebrate efficient strategies
  • Reasoning about area makes relationships pop so that the strategies are natural outcomes!

See this blog for a picture of Pam's model in this episode: https://www.mathisfigureoutable.com/blog/ep147
See Episodes 146 and 148 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

Show Notes Transcript

The area model can do more than build Spatial Reasoning. It can help make sense of the major multiplication and division strategies! In this episode Pam and Kim use a Problem String to show how reasoning about area can build more sophisticated and efficient multiplicative relationships.
Talking Points:

  • Multiplication is about grouping groups
  • The area model can help make sense of multiplicative relationships and properties
  • Choose and celebrate efficient strategies
  • Reasoning about area makes relationships pop so that the strategies are natural outcomes!

See this blog for a picture of Pam's model in this episode: https://www.mathisfigureoutable.com/blog/ep147
See Episodes 146 and 148 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 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 particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim:

So, last week, we started a series on the area model, and if you missed last week's episode, you'll definitely want to go back and take a listen because Pam shared why the area model is such a great tool, and also what we see people are doing when they're doing it just a little bit wrong. So, this week, let's dive. Just a little bit. Let's dive a little bit more deeply into understanding the area model.

Pam:

Yeah, and we're really not trying to call anybody out. What we're trying to do is help everybody because I think there's a lot of really good people out there that are trying to use the area model because they're sort of being told to or whatever. And if we can help everybody understand why and the power we can get out of the area model. Bam! Then, all of a sudden, it becomes a thing that you want to use, and you're happy, and it's helping you reason more and more like a mathematician. Alright, so the area model is spectacular to build spatial reasoning. It's also fantastic to develop chunking the chunks or grouping the groups. I heard Cathy Fosnot say once that, "Multiplication is all about thinking about groups and the numbers in each group, but we want kids really quickly to realize that we're going to group those groups." So, we're not just going to count each group at a time. We're not going to add, you know, if we're talking about seven-eighths. We're not going to add 8, and 8 and 8... We might begin that way as we think about multiplication, but then we want to group those groups. We want to chunk the chunks of area. That's why the area model comes in so handy because we're not just grouping the groups, but if we can line those groups up in an array, then we can think about chunking that area. And once we chunk those chunks, then we're really building that spatial reasoning. And we also can help make sense of the major multiplication and division strategies at the same time as we're learning area. So, it's brilliant. It's not just, "Hey, for these numbers in this problem, do I multiply? Do I add?" It's like actually feeling the relationships because we've helped build them with this area model, feeling what area means, having intuition for the square unit of measurement, and how that comes about from these dimensions. All of that stuff can come to play, and also helping make sense of strategy. So, we can build all of those things at the same time, making sense of strategy. Bam! We are going to do that today. Let's experience that today on today's podcast. So, Kim.

Kim:

Okay.

Pam:

I am going to give you a series of problems. We call that a Problem String. And I'm going to make your thinking visible. I mean, I'm going to draw it on my paper. I'm going to say out loud what I'm drawing on my paper, and then...

Kim:

Okay.

Pam:

And then, we'll try. We'll do our best. (unclear).

Kim:

Okay.

Pam:

I need to grab a piece of paper, so hang on just a second.

Kim:

And so, we are actually going to post this, right? We'll post a picture of what you draw afterwards, so that people can maybe listen to us talk, and they can draw their own based on what we're saying, and then maybe even compare it, if they'd like, to what we post.

Pam:

Ooh, nice, nice. Yes, yes, yes. And we'd love to see that on social media, and we can all learn from each other. Cool. Alright, I just clicked my pen open. Pen, Kim, Pen, pen. It's all about the pen. Pen. And here we go. Ready?

Kim:

Yep.

Pam:

First problem of the string. Kim, what is 15 times 14.

Kim:

Fifteen times 14? Do you want what I think about?

Pam:

Oh, yeah.

Kim:

Or do you want me to...

Pam:

No. Well, maybe for this problem. For this problem, go and tell me what you think about, but then I'm going to ask you what we would expect to see in a classroom. Because, listeners, if we just had Kim do this Problem String, it would not be typically what you see in a classroom, so we're going to kind of talk about what you would typically see. Kim's not typical.

Kim:

Okay.

Pam:

I don't know if you know that, Kim. (unclear). For sure she's not like a typical student learning multiplication. So, Kim, go ahead and tell us what you do for 15 times 14, and then I'll ask you what you would see in a classroom.

Kim:

Okay, so the first thing I thought about was just 15 times 10, which is 150.

Pam:

Cool.

Kim:

And then, 15 times 4.

Pam:

I'm going to pause you for just like it there. So, as you say that, I'm going to go ahead and draw a length. So, a vertical line on my paper. I label that 15. And then, you said 15 times 10, and then you said 4. So, I'm going to go and draw the 14. Not quite as long as the 15. I'm in pen here, so let's hope this... It looks pretty square. It's supposed to be a little bit longer, but you know 15 by 14 is pretty square, right? I mean, compared. But it should be a little bit deeper than is kind of sideways there. Anyway, and then, you said... You said 15 by 10, right?

Kim:

Yep.

Pam:

So, I've drawn kind of way over there like 10 ish. And then, you said 15 by 4, so that's a tiny length. So, I now have a rectangle on my paper that looks pretty square-ish, but it's a rectangle that is cut once. And it's a vertical line that's cut about... I was going to say two-thirds, but that's not right. Four-fifths. It's about ten-fourteenths of the way. Is that five-sevenths? It's about five-sevenths of the way along the line which labeled 10. And then, the other length there is labeled 4. Okay, and I think you were about to tell me what those chunks were?

Kim:

Yeah, 15 times 10 is 150.

Pam:

Mmhmm.

Kim:

And then, 15 times 4 is 60.

Pam:

How do you know that?

Kim:

Because there are four 15 minute chunks on a clock, four 15 minute chunks in an hour.

Pam:

Nice. And I would have thought about two 15's, and then doubled 30, but... Okay, cool. And then, so what is 150 and 60? Those two areas together? What's that?

Kim:

Two hundred ten.

Pam:

That's 210. So, you're saying that 15 times 14 is 210?

Kim:

Yep.

Pam:

Alright. So, the first problem 15 times 14 was 210, and you share that strategy. For this particular string, I might try to see if anybody did anything different. So, I might say because I'm looking for a specific different strategy. I'll tell you why in just a second. But I might say, "Hey, did anybody think about breaking up the 15 instead? Anybody think about? Like, Kim broke up the 14. Anybody think about breaking up a 15?" So, Kim, I'll just. You didn't, but could you? Could you break up the 15? Yeah. So, I... What would that look like?

Kim:

Break up the 15?

Pam:

Mmhmm.

Kim:

Yeah, so then I would think about 10 times 14.

Pam:

Okay.

Kim:

Which is 140.

Pam:

Mmhmm.

Kim:

And then, I would have the leftover 5 times 14.

Pam:

Okay, and so I've just drawn a rectangle that is hopefully the exact same size as the 15 by 14. I drew on the other ones not quite, but it's close. And then, I cut the 15 into 10 and 5. So, now I've drawn a horizontal line because I've cut it vertically, so I've got two vertical area, two vertical rectangles that I've cut that rectangle into. And I think you said 10 times 14. You already said that?

Kim:

Yep, that's 140.

Pam:

(unclear) And what's 5 times 14?

Kim:

Seventy.

Pam:

Do you know? You didn't memorize your 14's. You don't know.

Kim:

No, it's 10 times 14. So, 5 times 14 is half as much as that.

Pam:

Five is Half a Ten! Bam! Nice. And then, we could add the 140 and the 70 together. Is that also 210?

Kim:

Yep.

Pam:

Cool. Alright. So, I might model those two strategies. Notice, listeners, that what we didn't do... And I might have had students do it. But what I didn't choose to model as the teacher. I did not put on the board a rectangle that was 15 by 14 where I cut the 15 into 10 and 5 and the 14 into 10 and 4 at the same time.

Kim:

It will happen. Yeah, it will happen in your room, right?(unclear).

Pam:

(unclear). Yeah, and I'm going to celebrate student thinking by letting them do that. I'm going to let them solve it in a way that makes sense to them. I might walk around as I'm circulating, going, "Mmhmm. That makes sense to me. That makes sense to me." But I'm going to choose to model on the board, to celebrate on the board those students who did a little bit more efficiently because I want to nudge the math forward. I want to build in students this idea that we're looking for more efficient, we're looking for bigger, fewer chunks. So, that's a way to do that is to have those shared. I might even say to students, "I saw a lot of you cut it into those place value chunks. Notice up here. Could you? Do you have access to only cutting it into two chunks? Huh. Do you want your brain to do that next time? Okay." That's a great growth mindset message to send in that moment. Next problem. Alright, next problem is 30 times 7. Now, Kim, don't tell me what you would actually do.

Kim:

Okay.

Pam:

Tell me what you might expect if we were doing this in like a... I don't know. End of the year third grade, or fourth grade, or fifth grade classroom. What would you expect?

Kim:

Okay. Like we talked about before, you might have some kids that are making quite a few chunks, so conceivably you could have a kid who's cutting the 30 into some 10's. Would probably not be something I acknowledged. But you might have a kid who says, "I know 3 times 7 is 21, so then 30 times 7 is going to be 210.

Pam:

Nice. Yeah. And so, I can think about 30 times 7 as 3 times 7 times 10. 10 of those 3 times 7's. Now, for this particular Problem String, I'm not going to play on that idea too much. We do want to do work on that for sure. I would have presumed we would have already done some of that work, if I'm doing the particular Problem String that we're going to do now. So, I might call on a student who says that, and I might represent that on the board...I'm on a paper right now...as writing 3 times 7 times 10, and then. So, you know, what is that? What is 3 times 7 times 10? Go ahead, Kim.

Kim:

Two hundred ten.

Pam:

That's 210. So, I might just do that. I'm not going to do a whole lot of representing the 3 by 7's and finding 10 of them. That is work to do, but we would have done that before I do this particular Problem String.

Kim:

Yeah.

Pam:

So, then, I might say... In fact, if I was in a classroom, I might have said, "Hey, 30 times 7. Before you work that out, can somebody just help me draw it? Like, what would it look like?" Now, remember, I've already got this 15 by 14 on the board, so I'm going to be kind of pointing to that when I say that. I'm going to go, "Hey, you know like, based on this rectangle I already have up here, what would the 30 by 7 look like?" And I'm going to try to call on somebody who's looking like, "Yeah, I could describe that." Kim, how would you describe what the 30 by 7 would look like?

Kim:

So, I'm thinking about the dimension that you have. I'm sorry, I don't remember which way you did the 15. But where you had a dimension of 15, then I'm going to compare 30, and I'm going to make it twice as long on that dimension. And we're you had the 14, now we have 7, so it's going to be half as wide if you did it that way.

Pam:

Yeah, I did.

Kim:

So, twice as much one dimension, half the other dimension.

Pam:

Yeah. And because the problem was 15 times 14, the number of rows by the number of columns, I'm going to stay consistent that way. So, I did do the 15 was the number of rows by the 14 was the number of columns. So, this time, I did sort of do two of those 15's, so now, I have a 30 deep. And then by half as wide, 7. So, now I've drawn a 30 by 7. That is compared somewhat to that rectangle that I had above. I have 30 by 7. And then, I'll say to students, "Ah, and then you thought about it as 3 times 7 times 10." Or I might have had students, like you said, think about it as 10 times 7, and then double that to get 20 times 7, and then add back the 10 times 7 to get 30 times 7, I definitely might have that happen. I'm going to allow that to happen. I celebrate their thinking by allowing that to happen. "Yay, good job." But I'm not going to put that on the board because we're moving towards more efficiency. I'm trying to move the math forward, so I'm just going to represent the 30 by 7, put the 210 area in the middle of that rectangle. And then, I'm going to watch, and I'm going to look, and I'm going to see if anybody's noticing that we had two products in a row that were 210. So, I might even say something. I might even look at the board, and I might go,"210... Huh. Hmm. Oh, well, I'm sure that's just coincidence." And then, I might move on. I'll do something to (unclear). Now, I'll do that if nobody else said something. So, I might have had a student say, "What? Hey, it was the same!" And I might go,"It is? Oh, I'm sure that's just coincidence." So, I'm going to have it come up. I'm going to kind of, you know like, jokingly sort of discount it a little bit, and then I'm going to give the next problem. So, I'm allowing the opportunity for students to notice the products are the same and start to ruminate on that, start to wonder about that. Is there something about those products being the same that might kind of influence the way I see relationships from now on? Notice, that another move I made was specifically saying based on this 15 by 14, how would I draw the 30 by 7? And I'm looking. I'm going to pull out language, looking for students to say. I'm going to pull out language. Like, "Oh, I doubled the 15. And if you have the 14, then..." Okay, cool. Next problem. So, then, Kim. I might ask the problem 18 times 35. What might... Don't tell me what you would do.

Kim:

Okay.

Pam:

What might you expect to see students doing?

Kim:

You know, I hope one of the strategies that I'd see students do is consider 20 times 35. So, instead of 18, I would hope they would say, "I've drawn an array of 20, and then the other is 35. And so, 20 times 35 is 700. 2 times 35 is 70. So, 20 times 35 will be 700." But then there's too many 35's, 2 too many. So, then, they could, on their area model, get rid of two 35's, which is 70.

Pam:

Cool. So, I've done a rectangle that is 20 by 35.

Kim:

Mmhmm.

Pam:

And at the very bottom, I've drawn this tiny little bit that represents 2 by 35, and I've written 70 as the area. And then, I'm kind of crossing out that area a little bit.

Kim:

Yep.

Pam:

And so, you had said that 20 by 35 was what again? That whole area?

Kim:

Seven hundred.

Pam:

Seven hundred. So, I kind of have off to the side, I've sort of represented that whole area was 700. So, 700 minus that 70 would leave you with?

Kim:

Six hundred thirty.

Pam:

You think it might be 630. Cool. And I would absolutely think that we would have something like that happening in class. Now, unfortunately, I'm writing in pen. So, I might for this next one wish that I'd been writing in pencil. Kim's like,"See, told you."

Kim:

(unclear).

Pam:

On the board, I would have had a dry erase marker to maybe do something for this next problem. If I have a student at this point that is sort of thinking about... Well, let me backup because I don't want to kind of... Yeah, I'm going to move on, and then I'll tell you in a minute what I was about to say. So, the next problem... So, I'm going to have students share, probably one strategy for 18 times 35. I'll represent that. I'll make that thinking visible. Next problem. 9 times 70. And then, I'll see, "Hey, did anybody do something similar to the 30 times 7 where you did 3 times 7 times 10? Did anybody think about 9 times 7 times 10?" And I'll say as little of that as possible, try to pull that out, and then I'll write on the board 9 times 7 times 10. And I'll ask a student, and they'll say "630". And then, I'll be like, "Hey, so I wonder. Could we draw that 9 by 70? Let's draw that. What would that look like?" And remember, I've got that 18 by 35 above it. This is the point where I might have erased some of that Over strategy, so I only have the 18 by 35, not the extra 20 kind of thing, but. So, Kim, what would we expect a kid to say, to draw that 9 by 70.

Kim:

So, you drew the 18 dimension, and this time it's going to be only half as long, so that it represents 9. And then, on the dimension that's 35, you're going to want to make sure that dimension is twice as long to go from 35 to 70.

Pam:

And I don't know if anybody cares, but when I do that twice as long thing, I always like sort of draw the one length, and then kind of leave a little tiny space, and then draw the next. So, you can almost sort of see the Doubling because I've just kind of left a little tiny. If you look on the show notes where we'll post this, you'll see that kind of little space that I drew there. And so, now I've got a 9 by 70 on the board that is sort of where I've halved the 18 to 9, and I've doubled the 35 to get 70. And then, in the middle of that for area, I've written 630. Then, I'm going to stand back, and I'm going to go, "Huh. What are you noticing? And why?" And then, I'm going to try to elicit from students things like, "Well, I can see that if you cut that rectangle in half right there, and you kind of scooted that half over, you would create that new rectangle. And students will absolutely say that. We've done this how many times, Kim? They'll absolutely. And they'll go from both rectangles. So, for example, if I'm on that 18 by 35, students will say, "Well, if I cut that 18 in half..." and I'll draw a dotted line on that rectangle. And if I... In fact, maybe on my paper right here. I'm going to redraw that 18 by 35, and I'm going to cut it in half to get that 9. And then, I'm going to say, "If I took that half, and I kind of put it up over next to it to the right, I would create that 9 by 70" Oh, sure enough, I didn't lose any area. If you cut the rectangle in half and move that half over. And then, I would probably do the same thing up on that 15 by 14, where we'd cut it in half the other direction and kind of move it over. We could see that we could create that new rectangle and not lose any area. Huh. So, we've created an equivalent problem that's easier to solve. Sweet. "Well, class, that seems really interesting. I wonder if that can help you with this problem?" How about 4.5, or 4 and 5/10, times 16. Is there some way? Like, if you drew that rectangle is there some way that you could Double one dimension, Halve the other, and not lose any area and create an equivalent problem that's easier to solve? So, listeners, pause, pause, pause. Go solve that, listeners, see what you can do. Can you take 4.5 times 16? Double one dimension, Halve the other. Or Halve one dimension, Double the other. Move that chunk around. Don't lose any area. Create an equivalent problem that's easier to solve. So, Kim, do you want to that? What are you thinking?

Kim:

Yeah. The first thing I want to do is consider which number is worth halving and which number is worth doubling to make it nicer. The point is to think of something equivalent that's easier to solve. So, 4.5 does not look all that exciting to halve, although it could. So, I'm going to double 4.5 to get 9, and I'm going to halve 16 to get 8. So, then I know that 9 times 8 is 72.

Pam:

Nice. Nice. So, you can solve for 4 and a 1/2 times 16 by thinking about 9 times 8. And you're thinking that both of those have the product of 72. Super cool. And I could draw that 4.5 by 16. And then, I could double that 4.5 dimension and halve 16. And sure enough, we wouldn't lose any area. Ya'll, this is an example of how we can help build spatial reasoning, build the distributive property, build this associative property. We've got lots of properties happening here, but maybe most importantly help students really reason using area where they're really thinking about, they're using that area model to not only build area but to build a sense of why you can Double and Halve. Because ya'll, you can just tell students, "Oh, hey. Double, Halve. Here's a strategy. Memorize it. Go." And they'll just... It's another procedure for them to mix up and mess up. But if we develop the relationships, and area, and spatial reasoning, and set the stage for how the associative property is at play, and we get students ready to triple in third, or quadruple in fourth, or even when we get to the middle school, multiply by a number and multiply by its reciprocal, then we've got so much more happening, and math is so much more connected, and it's Real Math where kids are really building those relationships in their heads. All the things. Whoa! Super cool. It reminds me of some comments that I've been reading lately in our Building Powerful Multiplication workshop. So, we have a message board, and I'm in there responding to comments all the time. Where it's so interesting that there's usually a part in the workshop where there's several little sections where the comments all of a sudden start to go, "Oh, like the relationships are just happening." Like, "I'm starting to look at problems differently." "All of a sudden, my brain is pinging in different ways." Because they've actually built relationships, they have actually developed the way they're thinking, and the strategies become natural outcomes. They're just using what they own to solve problems. And it's so stinking cool.

Kim:

Yeah, that's really exciting. So, don't forget that we're going to post a picture right, Pam, of the board, the sketches that you just drew of the area model. And we would love it if you listeners would compare it to what you were drawing or what you were picturing for this string, and then share with us on social media.

Pam:

Absolutely. Ya'll, thanks for tuning in, and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit Math is Figure-Out-Able.com. Let's keep spreading the word that Math is Figure-Out-Able.