Ep 97: If Not the Multiplication Algorithm, Then What?

Kim Montague  00:01

Hey fellow mathematicians. Welcome to the podcast. Where Math Figure-Out-Able. I'm Pam.  And I'm Kim. 

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. But it's about thinking and reasoning, about creating and using mental relationships. We take a strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question: If not algorithms and step by step procedures, then what?

Kim Montague  00:38

Alright, y'all, we're in part four of our series, where we answer the question: If not algorithms, then what? We've already tackled addition and subtraction, and today we're going to tackle multiplication, what are the major strategies that we're using students, and anyone really need to know. Because if they own these five major multiplication strategies, then we can throw any multiplication problem that's reasonable to solve without a calculator, and they will be fine. They will be able to solve it as efficiently as a traditional algorithm and most of the time more efficiently than the traditional algorithm. But even more important than that, they'll be developing Multiplicative Reasoning. That is our goal: we want to develop reasoners, not just answer getters. Okay, so the five major multiplication strategies are: Smart Partial Products, Over/Under, 5 is half a 10, Double/Half, and Quarters. Alright, you got them. Okay. Thank you for joining us. I know we made that joke last time, but I'm gonna keep making it a little bit. Because we really want to emphasize it's not about the names. And it's not about us telling you these strategies. Now that we've sort of laid them out, now you need experience, your students need experience to develop these relationships actually create the relationships in your heads that these strategies are based on so that the strategies become natural outcomes. They become natural inclinations. As you look at numbers, you're like, "Oh, yeah, like this pings for me. And so now I'm going to use this strategy." And that's all about developing that reasoning, that level of reasoning that we're really after, not just answer getting. Alright.  Go for it. 

Pam Harris  02:25

One of us say 'alright' again. Alright, okay. Okay. Alright. Alright. (chuckles) Ready? Begin. 

Kim Montague  02:29

Okay, this week, we're going to tackle these strategies with the problem: 25 times 18. 

Pam Harris  02:36

Alright.

Kim Montague  02:36

Which I just wrote down with my pencil. With my pen. Pen, Kim. Pen, Okay. 25 times 18. We should do a listener's survey. How many of you write in pencil? How many write in pen? And then give all the people that do it correctly with pen...just kidding. Send them a sticker. Maths is Figure-Out-Able.  There you go. 

Pam Harris  02:57

International, 'maths'... Alright, 25 times 18. 

Kim Montague  02:59

Yep. 

Pam Harris  03:00

So similar to addition, a first strategy that we need to develop in students is kind of the place value splitting strategy. And we typically call that with multiplication, we typically call that Partial Products. In addition, we call it Partial Sums. You may have noticed, if you listened to the last episode, that in subtraction, we didn't really talk about partial differences. We don't find partial differences to be all that helpful. With students, it's okay if students do it, we support them, but we don't actively promote constructing with students. However, with addition, Partial Sums is important with multiplication, Partial Products is going to be important. So that is an important start. But oh, it's such just a start. That then we need to move and build the next five, we need to build on that. So what is Partial Products, that's when I look at thinking about 20. Again, it's sort of a place value splitting thing. I looked at 25 as 20 and 5, and then look at 18 as 10 and 8. And then I look at all those partial products. A typical way that we might see teachers help students with this, and we like this fine, is for students to think about an area model of a 25 by 18. And I'm drawing a rectangle right now. That's about 25 by 18, you might have noticed that I said I am drawing a 'rectangle', because to be clear, a 25 by 18 should be a rectangle not a square. Now a square is a rectangle, but it should be not a square rectangle. It should be a little longer than wider rectangle. It should be 25 long, 25 deep, and 18 across and that should be not look like a square. So if you've drawn a square, maybe we politely suggest that we're not developing very good spatial reasoning if we make something that should be long and not as wide. If we make it look square, no, no, no. It should look like a 25 by 18. Then, I would want to think about where I would sort of cut that 25 to make it 20 and 5. So I just sort of have a long part that's 20. And a much shorter part, oh, it's about a fourth of that 20, to be a 5. And then on the 18, I'm going to cut that as a 10, and 8. So the 10 should be longer than the 8. And now I have kind of four chunks. And those four chunks, I would want to think about that the 20 by 10 chunk. And that would be 200. And I would want to think about the 5 by 10 chunk. And that's 50. And I would want to think about the 20 by 8 chunk, I hope you sort of drawn these out, and I would think about that 20 by 8 chunk. And that's 160. And then don't want to think about that 5 by 8 chunk. And that's 40. Now I've got all these sort of partial products, all these little bits of area. And I would want to add those bits of area up. So if I add the 200 plus, oh golly, where do I go from here? I might, now it's actually kind of fun, because now you can kind of decide how you're going to add those up together. But to maybe make it easy on a podcast since you're just listening, I might add that 200 and the 160 to get 360. And then add the 40 to get 400. And then add the leftover 50 to get 450. I sort of added those up. And I'm thinking that maybe the product is 450, I think.

Kim Montague  06:21

Uh-hum. 

Pam Harris  06:22

Alright, cool. 

Kim Montague  06:23

Yeah. 

Pam Harris  06:24

That's like a baseline. That's a starting place. We're ending there. We're also not suggesting that the end is now, okay, now that you've kind of understood multiplication, now we're going to go do this algorithm that is so digit oriented, and so like these columns, and then I end up with all these numbers and columns I'm going to add. That could be very sort of step by step rote memory, no, no, no. Now that's a baseline. Now we want to go and develop these other five strategies. 

Kim Montague  06:49

Yeah. 

Pam Harris  06:50

So Kim, I think the first one that we would develop from there is?

Kim Montague  06:53

Smart Partial Products. So we're taking kids from these place value partial products into Smart Partial Products, which means that they don't have to break everything up into those same four place value chunks. So for 28 times 15.

Pam Harris  07:10

Eighteen.

Kim Montague  07:10

Sorry, 25 times 18.

Pam Harris  07:13

Did you just totally flip both numbers? That's hilarious. 

Kim Montague  07:15

I did. That would not be the same thing. 25 times 18. I think if I was thinking Smart Partial Products, I would think 25 times 10, which is 250. And then 25 times 8, which I know is 200. Whoa, you did that kind of fast. Do that again. Twenty-five times 10 from the 18. So I split up the 18 into 10 and 8. So 25 times 10, which is 250. And 25 times 8, which is 200. Can you talk me through how you do 25 times 8? Oh, because 25 is quarters. And so 25, eight quarters would be $2 which is 200 cents. So I could think about four quarters being 100. So eight quarters be 200. That'd be a way to do that.  Yep. 

Pam Harris  08:08

And you could also think about the fact that you just had 10 quarters, using that times 10 thing to be 250. 

Kim Montague  08:13

Yeah.

Pam Harris  08:14

But I need 2 less quarters than that. So that's also 200. There's a couple different ways to do that. So I noticed, Kim, that we did that Smart Partial Products, or maybe we call it 'clever partial products', that you broke up the 18 the same way I did into 10 and 8, but you left the 25 whole. And because you could think about ten 25s and eight 25s. You were being smart about it, you being clever about it, like maybe lazy about it.  Yeah. Let's only break it up as much as we have to, not into the same four chunks every time. But rather, ooh, like can I kinda be smart about it? 

Kim Montague  08:49

Yeah. 

Pam Harris  08:49

Nice. That's Smart Partial Products or clever partial products. Cool. Cool. Alright. 

Kim Montague  08:56

Oh. Next, you get to do one of my favorite strategies. Oh, you're gonna let me huh?  Yes, I will.

Pam Harris  09:02

Nice. Alright, let's do the Over/Under strategy, Over/Under. So for 25 times 18, I want to think a little bit over and then go a little under. So I'm going to think about 25 times 20. Because I can sort of think about 25 times 20. So I'm actually drawing another area model and I'm drawing a 25 by 20. And I'm saying to myself, well, I know that area of that whole thing is 500. And I did that by thinking about two 25s is 50. And then I scaled that up times 10 to be 500. I could have also thought about that as ten 25s is 250. But I need double that, so double 250 is 500. So a couple different ways to think about twenty 25s. But I only need eighteen 25s. So instead of a 25 by 20 I'm going to do 25 by 18 by getting rid of two 25s. So I have that 500 was the total and I'm getting rid of two 25d, which is 50. And 500 minus 50 gets me to that 450. 

Kim Montague  10:10

Nice. 

Pam Harris  10:11

So, okay, over? 

Kim Montague  10:13

Very good. 

Pam Harris  10:13

Hey, Kim, I'm curious, since you're the over queen, would you ever think about doing for 25 times 18, would you ever think about doing thirty 18s and getting rid of five 18s?

Kim Montague  10:26

I mean, they could, but I don't think I would. I can tell you why I wouldn't. I could think about 30, nah, none of those are nice for me. Thirty 18s? I don't know three 18s. I know two 18s. I do. And so I don't know why I do. But I do. 

Pam Harris  10:41

Is that 54? 

Kim Montague  10:42

Yes. Good

Pam Harris  10:44

Yeah. I'd think about it, though. But then also, once you had thirty 18s, then I'd have to figure out five 18s. 

Kim Montague  10:51

Yeah. 

Pam Harris  10:51

I can do that, cuz I could do ten 18s to get five 18s.

Kim Montague  10:54

That's the part that's a little yucky for me. It's the 90 have to think about.

Pam Harris  10:58

Yeah. So we could, but both of us are like, "Nah, probably not." 

Kim Montague  11:04

Yeah. 

Pam Harris  11:04

And that's the kind of choice we want to give kids. 

Kim Montague  11:07

Right? 

Pam Harris  11:07

We want kids to be able to do exactly what we just did. I mean, I could, but nah, there's a better one. In fact, honestly, neither of the two we've done so far are my favorite for this problem for 25 times 18. Or any of the ones we've done. Partial Products is not my favorite. I mean, I can. Smart Partial Products, I think he did a great one. I could. This Over/Under. I mean, I could, but not my fave, not for this problem. So what's the next strategy?

Kim Montague  11:34

Five is Half of 10. Alright, that's on you. I think.  Ah. I was hoping you would let me do the next one.

Pam Harris  11:40

Oh, okay. Well then I'll do 5 is Half of 10. Okay. So I'm gonna think about 25 times 18... 

Kim Montague  11:44

Okay

Pam Harris  11:45

..by thinking about, I'm not going to break up the 18 this time. I'm going to break up the 25 into 20 and 5. So I'm gonna think about twenty 18s. And I'm actually asking myself if I can do that on an area model or a ratio table, and I think I'm doing a ratio table. So I'm thinking about 18s. And I'm going to think about ten 18s is 180. So twenty 18s is 360. I could have thought about two 18s is 36. And then scale that up to get the twenty 18s  is 360. Either way I have twenty 18s. 

Kim Montague  12:15

Yeah. 

Pam Harris  12:16

But I did the 10 on purpose, because I knew I was going to try to get the 5. Does that make sense? So I could have done 2 to scale it to 20. But then I wouldn't have the 10 to get the 5. And you want me to do the strategy 5 is Half 10. So since I know ten 18s is 180, I now know five 18s is half of that 180. Since 5is half a 10, then five 18s would be half of 180, which is 90. Now I have twenty 18s being 360. Five 18s being 90. And so I can add the 20 and the 5 together to get 25. And the 360 and 90 together to get 450. And I totally just thought about angles when I did that 360 and 90 as angles to get the 450. Anyway, so 5 is Half of 10 is using this relationship that's so cool in our base 10 number system. That I can find 5 times anything by finding 10 times that thing and then cutting it in half. And you might be like, "Wow, why would you wanna go to that work?" Well, it's because 10 times things are so easy to find, right? In our base 10 number system, multiplying times 10 is so easy, because there's zero thing and we can talk more about that. But I can use it. I can use the idea that I can do the scale change, where I'm scaling by 10. I can shift things in the place value, or I can think about that and then just cut it in half to get times 5. Bam! I've got that times 5, and I can use that times 5, however I want to. This case just added it back to the 20. And then I've got 25 times 18. 

Kim Montague  13:49

Yep. 

Pam Harris  13:49

Cool. Alright. So what do we have so far? We've done Partial Products, which isn't one of our five then we did Smart Partial Products. We did Over/Under. We just did 5 is Half of 10. Before we go on Kim, if you don't mind, I want to just note that all four of the strategies we've talked about kind of the baseline partial products, and then the other three that are three of our favorite five. All of those are based on the distributive property of multiplication. So the distributive property multiplication says that I can sort of distribute, which means I can find these chunks of area. And then I can kind of chunk them and add them together. So that's the distributive property. Because the next two are based on the associative property.

Kim Montague  14:29

This is one of my favorites, right? And this one's... 

Pam Harris  14:32

Alright, Double and Half for us.

Kim Montague  14:33

This one the name, right? We get stuck on the name here, because... 

Pam Harris  14:36

For this problem, especially. 

Kim Montague  14:38

I think you've made some offers before that if we have a better name that encompasses all that this strategy entails that, you know, we definitely want to hear it. But for this problem, I would absolutely think about double half. Because I know that if I have, and I actually think about this one is 18 times 25. So is that legal? Can I...? 

Pam Harris  14:59

Use the commutative property? Sure. 

Kim Montague  15:00

Can I think eighteen 25s? Okay, so instead of eighteen 25s, I want to create an equivalent problem, that is 9 times 50. So instead of eighteen 25s, I need, if I want half as many of them that I need...Oh, I always say this wrong. 

Pam Harris  15:21

Twice as much? 

Kim Montague  15:22

I need twice as much in a group. I need half as many that's twice as big. 

Pam Harris  15:27

Yeah.

Kim Montague  15:28

So instead of 18 times 25, I want 9 times 50. Because I know that that's 450. You know, nine times 50. You can think about that? I do because I know 9 times 5. 

Pam Harris  15:40

That's 45, Scale that up by 10, that's 450. 

Kim Montague  15:43

Yeah, yep. 

Pam Harris  15:44

Or we could keep going. If you're thinking about nine 50s, you'd say,

Kim Montague  15:49

I could think about people, four and a half times 100. Oh, place value shift. And that's also 450. Four and a half times 100. That's kind of fun.  Yeah. 

Pam Harris  16:03

Like it. So Doubling and Halving, we could totally think about how Kim just said it. If I'm going to have half as many groups, the groups have to be twice as big. I can think about it the other direction. If I'm going to have, so if I was thinking about twenty-five 18s, but I could say, oh, I want 50. I want twice as many groups. But if I want to keep the same amount, then I got to have them be half as big. So instead of finding twenty-five 18s, I'm going to find fifty 9s, that's twice as many groups that are half as big, fifty 9s. It's just another way of thinking about that. But we could also look at an area model for doubling and halving. If I looked at an 18 by 25, area model, and I cut that 18 in half. So now I have a rectangle, that's a 9 by 25, and another 9 by 25, sort of piece, and I kind of shift that piece over next to it. So now I have a 9 by 25, next to a 9 by 25, that turns into a 9 by 50. And that 9 by 50 has the same area that I had before I cut it up. And so since I've the same area, then the products are equivalent. So cool. And this might spark in you a very similar to some strategies that we did in addition and subtraction, where by using this strategy, we've created an equivalent problem that's easier to solve. So in addition, when we do a little Give and Take, why do we give and take? Because we create an addition problem, it's easier to solve. In subtraction, we shift the distance. We think about the subtraction problem as distance or difference. And we shift that distance or difference on the number line and we create a subtraction problem that's easier to solve. With Doubling and Halving for multiplication, we move area around a little bit, and we create a rectangle with the same area or in other words, a multiplication problem that's easier to solve. We create an equivalent multiplication problem that's a little bit easier to solve. Those are cool strategies, when it's all about creating an equivalent problem that's easier to solve. We like those strategies. Very nice.

Kim Montague  18:09

Yeah. 

Pam Harris  18:11

I'm looking at the time, I think I have just a little bit of time. I'd like to describe a little bit how that's based on the associative property, if I may. 

Kim Montague  18:18

Sure. 

Pam Harris  18:18

So when I'm thinking about 25 times 18, I could also think about that as 25 times 2 times 9. And right now when I think about 25 times 18, a 2 times 9 is associated. So think about it, 2 times 9 is 18. But if I reassociate, that 2 instead think about 25 times 2, and then kind of leave that 9 hanging out there, 25 times 2 times 9. I'm associating the 25 times 3, and leaving the 9 out there. Now 25 times 2 is 50. And I still have that 9. And then I turn it into that 50 times 9. Bam! I've re-associated that 2. Instead of having it sort of in that 18, I'm pulling it out and I'm associating it with the 25. That's how you use the associative property. So cool, so stinking cool. 

Kim Montague  19:08

Alright.  Alright. Last strategy. Before we actually do the last strategy, I'm just gonna say we have these four major strategies. And when I got really serious about whoa, like, if we never teach kids, the algorithm will, like if I never use an algorithm - and really I was thinking about my own personal kids, because my own personal kids were the grand experiment. Like what if they never ever see a traditional algorithm? Will they be able to solve any problem that's reasonable to solve without a calculator with a strategy that we've developed? And I decided not quite with the four that we've just talked about. Golly, can I go over them: Smart Partial Product, Over/Under. Yeah, help me. Five is Half of 10 and then the last one. 

Pam Harris  19:52

With those four, would I be able to solve any of them? Is there another strategy that we need so that we can really solve any problem that's reasonable to solve without a calculator? And I decided to add this one last one in and I call it Using Quarters. Kim, am I doing this one? Are you doing to this one? 

Kim Montague  20:09

Yep. 

Pam Harris  20:09

Okay, for 25 times 18, I'm going to think about, I can think of it in two ways. But I'm going to think about it as 1/4 of 18. I'm gonna think of 25 is like 25%, 25 out of 100. And so if I think about 1/4 of 18, let's see 1/2 of 18 is 9. And so 1/4 of 18 would be 1/2 of 9 is 4.5. But that's like thinking about .25 or 25 hundredths times 18, right? A 1/4 of 18 is like .25 times 18 is 4.5. But I want 25 times 18. I want 100 times that. I don't want just .25 times 18, or 25 hundredths of 18 or 1/4 of 18. I want 25, not 25 hundredths. So I've got a scale that 4.5 times 100. Four point five times 100 is 450. Now, you might be like, "Whoa, why is it is that crazy work with fractions?" Well, it becomes really nice if the problem is actually like 75 times 18. Then I can say to myself, well, if 1/4 of 18 is something, is like 4.5, then can I think about three of those one fourths. Or I can think about if 1/4 of 18 is 4.5, then 25 times 18 is 450. Then I can think about three of them to get 75 times 18. So I can sort of think about three of those. So how are you thinking about three 4.5s or three 450s. I can think about 900. And then another 450 is what 1350.. So 75 times 18 now becomes 1350. And now I can think about seventy-six 18s by just adding one more. A little Over/Under, I just tack one more on there. I can just tack one more 18 on there. Is that 1368? I mean, bam, like I'm solving all sorts of crazy problems based on sort of this fractional relationship. And not only am I doing multiplication, but I'm also kind of practicing this idea of fractions, a fraction of a number. And I'm using a fraction as an operator, which is one of those meanings of fractions that is so important, fractions as an operator. So lots of relationships kind of come to play. It can become the brilliant work that we do in sort of fifth and sixth grade, as we want students to be grappling with all of those ideas. And you might be like, "Pam, why would I ever do that? I could just do these steps." I'm like because remember, our goal is to build multiplicative reasoners, is to build Multiplicative Reasoning in students. This is a brilliant way to do it, to have them play around with these relationships.

Kim Montague  22:55

Okay, do we have time for one more bonus strategy? Yes, as long as we do it quickly. We want the episode to stay manageable, right? Yeah. So one bonus strategy for...give me a grade. Give me age.

Pam Harris  23:12

I think it's sort of a middle school strategy. I think it's when we want kids really like we just said, building Multiplicative Reasoning. And so we've kind of got kids kind of multiplying, but now we want to really get multiplicative, Multiplicative Reasoning. 

Kim Montague  23:23

Yeah. Yeah. 

Pam Harris  23:24

And we call it - 

Kim Montague  23:25

So Flexible Factoring. 

Pam Harris  23:26

Flexible Factoring. Yeah.

Kim Montague  23:28

Alright, so I'm gonna let you go. Go quick: 25 times 18. Okay, so I would want to factor 25 into 5 times 5, And factor 18 into 2 times 9. And then I want to be flexible. Now that I have 5 times 5 times 2 times 9, I'm going to pull a 5 and a 2 together to create a 10. Because 5 times 2 is 10. What am I left with? I still have a 6 times 9. 5 times 9 is 45. And so I have 10 times 45 is 450. Bam! So Flexible Factoring is kind of advanced strategy. It's not one of our five main ones. But we do want middle school teachers to develop it after, where you factor the factors in the problem. And rearrange the factors using the associative property and put them together in clever ways to create 10s or create other numbers that are easy to multiply by, and then bam, you end up with another equivalent problem that's easier to solve. So that's also one of those really cool strategies. It's all about creating an equivalent problem that's easier to solve. Excellent. Okay, if you haven't yet gotten yourself this fabulous download. 

Pam Harris  24:30

Y'all, what are you waiting for? Come on! 

Kim Montague  24:32

Seriously. So much help. Alright, so it's got multiple strategies for each of the operations. You're going to find that download at mathisFigureOutAble.com/big. 

Pam Harris  24:42

Because it's big. 

Kim Montague  24:45

Get it today?

Pam Harris  24:46

Yeah, it is a big download. Alright. mathisFigureOutAble.com/big. Download that guy because if you want to learn more mathematics and refine your math teaching, so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able.