Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  00:09

And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:16

Because we know that algorithms are amazing historic achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:29

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:36

We invite you to join us to make math more figure-out-able.

Kim  00:41

Hi there. 

Pam  00:41

Hey.

Kim  00:42

I'm laughing because there's a note to you for time. Oh, where it

Pam  00:49

says "Be short. Don't make this one be too long." Except, you know, a couple weeks ago, I was in the frozen north of Alberta, and the teachers there drive farther. And they were like, "No, you could make the episodes longer." 

Kim  01:04

No! No, no, no.

Pam  01:05

Like most people were happy when we keep them snappy. Let's, let's snap today. Alright, we're snapping today. 

Kim  01:11

We're snapping. We're getting started. So, last week, we started talking about sequencing. And sequencing... We didn't talk about sequencing strings. We talked about sequencing different kinds of tasks together to help build foundations of number. And so, we talked about some young addition and subtraction last week.

Pam  01:32

Relationships. Yep, mmhm.

Kim  01:33

Mmhm. So, let's talk today about another area, and let's talk about multiplication, division.

Pam  01:40

Yeah. So, today we're going to focus on single-digit multiplication, division. Sort of kind of multiplication facts, and what does it mean to have early division. 

Kim  01:50

Yeah.

Pam  01:51

How do you build the foundation for those things? 

Kim  01:54

Yeah. 

Pam  01:54

Super important, yeah. Go ahead.

Kim  01:55

Yeah. So, if we say "foundation", I think almost everybody would say, "A foundational thing that kids need to know first before they can do anything else is they need to memorize their multiplication facts."

Pam  02:08

Yeah, "Because, kids, you can't do anything unless you have the building blocks, and so you've got to like start with the building blocks, and then go from there." That is very true if my goal is to get kids really good. Well, not really good. Just even good. If my goal is that kids are going to repeat the steps of the multiplication algorithm, then we are all really clear that they're going to do a lot of single-digit multiplication facts. Yeah. And we mentioned last week that for single-digit addition or single-digit subtraction, it would be super helpful if kids really owned the relationships. How do I say this? If I'm focused towards an algorithm, then kids need to know those single-digit facts because they're going to do them over and over in the addition and subtraction algorithm. But you know what? In those addition and subtraction algorithms, kids can often get away with counting.

Kim  02:58

Mmhm.

Pam  02:59

And it looks fast enough that teachers don't even notice that they're counting one by one in those algorithms, even though they should be developing... Sorry. Should be developing addition and subtraction additive relationships. Because they can often count, it's just single-digits that they're adding together in column after column, or subtracting in column after column, that they can kind of do that getting away with counting. But, ya'll, that is much harder to do in the multiplication algorithm. So, if you are focused towards getting kids to do 1-49, Odd of multiplication problems where they over and over, they're doing all of these single-digit multiplication facts, then yeah, it makes sense that you would say, "Go memorize those multiplication facts. Get them down because it's going to make everyone's life so much easier when you have to repeat all of those single-digit multiplications in the multiplication algorithm over and over and over until you die." 

Kim  03:50

Yeah.

Pam  03:50

Super true in the multiplication algorithm.

Kim  03:52

Yeah. So (unclear).

Pam  03:53

What if? What if? What if the multiplication division algorithms are not your goal? 

Kim  03:58

Yeah. So what do we believe... 

Pam  03:59

Change the story, yeah.

Kim  04:00

...are some foundational learnings for single-digit multiplication? We believe that doubling is a massively helpful strategy. So, you can double for 2, 4, and 8. So, twos are doubles, four is double double, eights or double double double. But not only that, if you know threes, then you can double those to get sixes. So, doubling is a hugely helpful strategy. 

Pam  04:26

And let me give you a specific while we're here. 

Kim  04:28

Yeah.

Pam  04:29

An often missed fact, a most missed fact. You know, if you test kids to see, out of all of those, often kids will miss 6 times 7. Like, that's a missed, an often missed, or most missed fact. 6 times 7. But if you know 3 times 7. Which a lot of kids know or they can find fairly quickly. 3 times 7. That's 21. Well, if three 7s is 21, bam, six 7s is double 21. 

Kim  04:53

Yeah.

Pam  04:54

Almost everybody can double 21. And if you can't, start doubling with kids, and pretty soon they'll double, you know. 21 pretty readily. So, 6 times 7 becomes a very figure-out-able fact if we've got that doubling down. Like you said, even eights. 8 times 7, I can double, double, double. Double 7 to get 14. Double 14 to get 28. Double 28 to get 56. Again, you might say, "Pam, kids can't double 28." And I'll say, "Start doubling. Like, getting kids..." That's what you're getting at right now, right? Is if we double with kids, then doubling it becomes this major relationship that is a thread that runs throughout mathematics. Mathy people play with doubles. Sorry for interrupting. 

Kim  05:34

No, it's okay. So, not only are doubling strategies helpful in single-digits, they're also useful in larger numbers as well. So, we're not just focusing on single-digit. Although that's what we're talking about today. The ones that we're going to mention are helpful outside of the single-digit number. So, another important foundation that we think kids can work with are making use of 10. So, times 10. Not only 4 times 10, but also times 10 can help you with times 5. So (unclear).

Pam  06:06

Yeah, you don't have to know your fives if you know your tens. 

Kim  06:08

Yeah. Yeah, yeah. 

Pam  06:09

Yeah, nice.

Kim  06:10

Also tens help with nines. So, you could subtract one group and get times 9. So, making use of 10 is super important. We also think an idea, foundational idea is partials. So, using chunks that you know. Which, again, helps outside of single-digit facts. We also think really foundational idea is the commutative property. So, within single-digit facts, we know that sevens are stupid. Sevens are often... 

Pam  06:40

Another often missed fact. Yeah.

Kim  06:42

Yeah, yeah.

Pam  06:42

Yeah. 

Kim  06:42

And so if you understand the commutative property, then you can think about a strategy for the other factor. So, those are some of our foundations (unclear). 

Pam  06:51

I just kind of threw out, right? Yeah. So, when I said 7 times 8, 8 times 7 is an often missed fact, so don't do 7 times 8. Do 8 times 7, and then you could double, double, double. That's just one strategy. You could also use 10, find times 10 and get rid of 2 of them to find 8 of anythings. But yeah. So, if you don't like times 7, or if you don't like 7 times something, then do the other factor times 7. And that's super, super helpful. Yeah, nicely done. So, if those are some of the major relationships that would build towards the major strategies for single-digit multiplication. But like you said a minute ago, also they grow up and they lead to the major strategies for a multi-digit multiplication. Which then all leads to the connections for division. Super cool. Kim, what are some of the types of tasks? We started today talking about sequencing tasks. What are some of the types of tasks we think about when we think about creating sequences of tasks?  Which

Kim  07:54

Yeah. So, just like we talked about last week, we believe that there are some messier, messing around, types of tasks that are important. We also think that there's some ways to focus on strategy and analyzing strategy. And we think that there are some ways to practice. So, often, we would start with a task. We would build in some Problem Strings, some games, routines. And then we'd land with some sort of practice that kids can use.

Pam  08:19

Yeah, nice. So, Kim, let's dive into a sample kind of sequence of tasks that we might do to help develop early multiplication and division, and kind of talk about the categories and how they fit together just to give everybody kind of an idea of what it could look like to do exactly what you just said. Like, what does it mean to start? Oh, go ahead. 

Kim  08:19

Oh, I was just going to say before you share a sequence that we've come up with. I just wanted to say that last week as you kind of talked through a sequence, that part of what I was thinking about is this is just much more fun, and engaging, and intriguing. Like, when I think of a sequence of a textbook that is like 4.1, 4.2, 4.3. It's like the same, same, same, same. And so, maybe, listeners, as you're thinking about the descriptions of the tasks that Pam is going to talk through, like picture what's happening with you and with kids and just the different feel that you take with these kinds of sequences.  Ice cream. 

Pam  09:24

Yeah. And I'll just pipe in the different kind of brain activity that's happening. That you're sort of engaging kids in different ways is then therefore engaging. Which is kind of a cool thing. So, you mentioned three kind of categories. Something that's a little bit more messy. If I wanted to have something that's a little more... Now, I don't mean "messy", as in unstructured, unsafe. The kids don't know what to do. They're fumbling around. I just say, "Go discover mathematics. Come back when you do." It's not that. It's the kids are very clear on their task. They're clear on what they're supposed to do, what their role is. They're clear on how they're supposed to engage with what they're doing. But then as they dive into that, there are several kind of ideas that they're playing with. There's several mathematical relationships that could come up for them, and they might focus one direction or another. And part of the teacher's job is to be very interactive, circulating with kids in like a problem like the following. What if we were to take more of a product approach and say, "Hey, instead of talking about the..." In fact, let me just back up a little bit. Often, when teachers are trying to help kids learn their single-digit multiplication facts, they'll say, "Alright, we're going to start with the twos, and we're going to do all the twos. Then we're going to do the threes, then the fours, then the fives." And I might be, you know, you might get a prize if you got all your sixes. Yay! You get the..." "...Ice cream sundae. The topping on the Sunday." Whatever. You get your your sevens. Yay, you get a spoon to eat your ice cream." Whatever. So, notice that that's like a factor approach, but we're going to take a product approach to be a little messier. Instead of just like hammering one factor, what if we were to say let's focus on, say, the number 12. And if Grandma is going to build a garden, and she's going to have a square foot garden, and each plant takes a square foot. And so, we give kids tiles, square tiles, and we say, "Hey, here's 12 tiles. What could grandma's rectangular garden look like?" And then we're going to have kids, what, create a 1 by 12 and maybe a 2 by 6. Another kid might create a 6 by 2. And as long as it's a rectangle, then we might say, "Well..." Like, I'm circulating around as a teacher, and I see a rectangle, I might say, "How many can you find? Keep going. And how do you know if you have them all?" That might sound familiar with last week's task where we threw out an addition situation like that, where we looking at like lots of addends. Well, this time we want to look at all the factors. What are the factor pairs that come out of 12? Once kids are kind of solid on that. And, again, we want to help them be strategic about how they are organizing, so that they know if they have them all. That's a little bit of an extension task where other kids are just like thinking about, well, connecting area, and perimeter, and dimensions to factors and products. All of that's happening while they're also connecting that there are multiple iterations. I can make multiple rectangles for 12. Once we've got those kind of listed out. We've been sort of systematic, so that we know if we have them all. Then we could say, "Alright, now I'm going to give you in partners. I'm going to give you other numbers." And I might give a certain partnership numbers like 9 and 13. Go. And I might give another partner 24 and 25. Go. And I might give another set of partners 36 and 33. Go. And I might give part... Let me do one more. I might not give a partnership 13 and 17. Why not, Kim? Why would I not give a partnership 13 and 17.

Kim  12:58

Because they're both prime. 

Pam  12:59

They're both prime, and so the only rectangles they'll come up with are 1 by 13 and a 1 by 17 or the community property there. And so, I'm probably going to give... If I give them a prime, then I'm probably going to give them really nice composite. But I also might naturally differentiate for kids by determining which partners I give them. If I've got kids that I know are going, going, going, and they really need a challenge, I might give them 36 because there's a lot of factor pairs for 36. I also might give them... Trying to think. What would be a (unclear).

Kim  13:29

You can give me 8. 

Pam  13:32

You want me to give you 8. 

Kim  13:32

I'll take 8. Yeah.

Pam  13:34

You want? 

Kim  13:34

You can have 36. 

Pam  13:36

I was thinking you want 7 or 5. 

Kim  13:39

I'll take it. I'll take it.

Pam  13:41

But I might give... You know, a group, I might give them a prime and a composite. Oh, by the way, in Canada, it's composite. 

Kim  13:47

Oh. (unclear).

Pam  13:48

That's cracking me up a little bit. Yeah. I love regionalisms. I hope you never lose them. I think it's so cool. Anyway, so I might be judicious about the kinds of numbers I give different groups. Not because I'm dumbing, trying to dumb down for some kids. It's I'm trying to give them just on the edge of their zone of proximal development, just enough to keep them grappling, but not getting too frustrated. 

Kim  14:08

Yeah. 

Pam  14:08

And I also want to really challenge groups that are ready on the edge of their zone of proximal development. 

Kim  14:13

Yep.

Pam  14:13

But then, Kim, then I want to have kids make posters for each of those where they've put the arrays they've created, those rectangular gardens they created. No,  we're just, at that point, drawing closed arrays. They're cutting out grid paper. But then I want to put those up on the wall. So, now I've got the numbers 1-36 on the wall, each having a poster representing the different arrays that we could create. Now, we're having a conversation about patterns, where they can literally go... Or I'll ask them, "What do you see? Why do some numbers only have a one by? Which numbers are they? Hey, do you know mathematicians call those prime." How about if we have some numbers that have... Oh, sorry. 1 by. If they have a 1 by, they could be odd. If they only have a 1 by then they're prime, right? 

Kim  15:04

Yes.  Yeah. 

Pam  15:04

Oh, everything has a 1 by. What am I saying? Help me. Even! Even are going to have a 2 by.  Odd won't have a 2 by. There we go. Ah! 

Kim  15:14

But in that, kids...

Pam  15:15

Clean that up for me, Kim. Say that right. Say that better for me.

Kim  15:18

Evens will have a 2 by. But kids will find that pattern. This is not you telling all these things. You're putting them up in front of kids that they've made, and you're sitting back going, "Notice. What do you notice?" And a kid will say, "Wait a second. Some have a 2 by and some don't." "Why? What do you notice?"

Pam  15:39

And then we can drop in the vocabulary just in time. "Oh, you're noticing that there's some even numbers and the odd ones don't have a 2 by Yeah, that's what we call those. Yeah. And you're noticing that there are some numbers that only have a 1 by. They don't have anything else. Oh, yeah. Mathematicians call those prime. And some people are really interested in prime numbers. Do you see a pattern? Like, is it every certain numbers? There's not a pattern for every. Whoa. Do you know there's not a pattern? Like, people are still looking for patterns for primes." And then students will notice square numbers and which numbers had squares. Is there a pattern for square numbers? Lots of things. Again, like nicely said. That's an example of what we mean by messy. It's messy because kids can notice things and we can pull those patterns together, but they can notice different patterns. That's the messy part of it, right? So, there's an example of kind of a messy thing. Another example of a messy thing could be where we want to get kids thinking about skip counting on an open number line, where we could have them thinking about a race. And if we have a race that's so long, and we're going to put aid stations at certain points, then if it's every... Say, if it's a 70 kilometer race, and we're going to put race stations every 7 kilometers, like where would we put them? And we could have kids really be thinking about multiples on a number line or what it looks like to have sort of skip counting on a double open number line. We could ask them certain questions about where they would go and how they know. We could also build a ratio table with kids by literally starting to plan a party. And we could say, "Well, hey, if we're planning a party and these are the things that we need, fill in the table for different numbers of things that we need at the party." And kids could ask and answer questions about that party planning all around the idea of sort of building a different model, the model of a ratio table. Notice, that in this messy sort of category, I'm mentioning three major models. One of them was the area model, where we gave kids tiles and we're having them create the Grandma's garden. Another one is an open number line with the race stations. And another one is the ratio table where we're planning parties. So, often we like to build models by having kids in situations where the model is is helpful to really make sense of what's happening in the situation. And that model starts to emerge as a math model, mathematical model that we can use to to build more relationships. 

Kim  18:06

Yeah. Also in this messy time, messing around time that you're building a model, but you also are steeped in a context, and you're building some things that you can come back to all throughout the year.

Pam  18:19

Lean back on. Yeah. When kids want to do something, say, on a ratio table, you could be like, "Ooh, wait. If it's the number of cakes, do you have to..." Let me give a better example. If I've got bags of cups for the party, and I have so many cups. If I add the bags, do I just... If I had one bag, do I just add 1 cup? Nah, I got to add all the cups in the bag. So, you can reach back to that context to help kids make sense of the model when the context isn't there if you've developed a really nice context. Yeah, that's nicely said. So, those are some examples of things that you could do that are kind of messier. But we would want to sequence those with tasks in that category where we're building strategies. When we're building strategies, that's where Problem Strings come in where we're going to build really the relationships that lead to strategies becoming natural outcomes. But we can also do things like put a multiplication chart in front of students. Now, not as a retrieval device where I say, "Oh, you can't memorize your multiplication facts, so I'm just going to give you this chart, and you just go drag your fingers to the thing, and you just retrieve that fact." Ya'll, for a retrieval device, how about if we just give them a calculator? We don't love retrieval devices. We'd much rather have kids build their multiplicative reasoning. One way we can do that is putting a multiplication chart in front of them and asking kids what patterns do you see? And when they see patterns on a multiplication table, on that chart, then we can say, "Huh. Like, you're thinking that if we look at this row, it's all 0. Why would that be? Oh, because you're multiplying by 0. And if we look at this... Kim, what's another pattern on a multiplication chart that we might want kids to pull out?

Kim  20:02

The twos. That they're going to all end with an even number. 

Pam  20:06

Nice, nice. One more. What's one more cool pattern.

Kim  20:10

Fives are going to end in a 5 or a 0. 

Pam  20:14

And they're also going to be half of the 10 row. Yeah. So, if we look at the 10 row or the 10 column, the fives are going to be exactly half of those. So, lovely charts that are... Lovely charts. Lovely patterns that can come out on multiplication charts. We can also then use those three models that we began to develop in the messier tasks. We can put student strategies on those models and smudge out part of their work, and then say to students, "Hey. Like, fill in the smudges. And now that you filled in the smudges, what was this kid doing to solve that problem? Like, in this ratio table, you just filled in the smudges. Here was the problem they were solving. What relationships did they? Oh, they multiplied by 10 and cut it and have to get 5. Oh, they were. Oh, and this kid? What did this kid do? Oh, they multiplied by 10 and cut it in half to get 5. Huh. What did this kid do? Oh, they multiplied by 10 and cut it in half to get. So, you don't have to know your your fives, if you know your tens. You could just... Oh, well, that seems like a helpful strategy." How are we starting to develop the foundation for that strategy? We've given kids a model. We've smudged out part of the work. And then we're analyzing what the kid was doing in that particular model. Another example could be an open array. We've now got an open array where we smudged out work, and we might see a something by 10 that's cut in half to that something by 5. And when they fill in those blanks, they're like, "Whoa! Like, I can model that strategy," in the ratio table I just talked about. But I could also represent that in this open array. So, same strategy on different models. All developing kind of these undercurrent relationships that build the foundations for strategies. When we've built one of these major strategies, or at least when we've started building it, we've put some words around it, we can create anchor charts for those strategies. So, I would want to have done some of this messy stuff and some of these smudge problems to analyze, looking at the patterns of the multiplication table, doing Problem Strings to get the strategies to be happening. And then once we've put words to it, now I'm creating a strategy anchor chart that says you don't have to know your fives if you know your tens. Like Five is Half of Ten. And I could represent that on a couple of different models. Hang that on the wall. Now, when kids are messing around, they're trying to make those single-digit facts and bigger numbers. When they're trying to make those more automatic, they could refer to that anchor chart. That's not the only reason we create anchor chart. We create anchor charts to anchor the learning. By putting words to the relationships that are happening, kids get more clarity around it. But now, also, we can refer to it because it's on the wall. We can continue to get to cinch those ideas. So, those are some ideas of things that we can do to help kids build foundations for strategies by analyzing patterns and strategies. The third category that you talked about, Kim, is we do want kids to practice.

Kim  23:12

Mmhm. 

Pam  23:12

We just don't want to be mindless practice. It's not practice to rote memorize. It's not practice with a song, or a rhyme, or a rap. It's practicing really in maybe in a few ways. So, here's maybe our favorite way. We really like the... Well, two kinds of practice, two kinds of cards to practice single-digit multiplication facts. One of them is similar to the cards that I talked about in last week's episode for single-digit addition facts. We call them hint cards where it looks kind of like a flash card. And, ideally, you would interview kids, and you would say, "Here's a fact 7 times 8." And when the kid says, "Mmm," and they have to fuss too long, and they're doing something unsophisticated to find it, then I put that in the pile, "Facts I don't know yet". But if I show you 3 times 4, and the kid just is like, "12," and like they know it pretty readily, I put that in the facts they know. Well, then we just take that smaller deck of cards that is the, "Facts we don't know yet," and we help kids come up with good hints. We look to that anchor chart that had the relationships on it that has those strategies we started to build, and we're like, "What would be a good strategy to use to figure this one that you don't know?" And now, when I flash you those hint cards, now you can use that hint just enough, not too much. Don't give it away. Don't make it too laborious that we have to like spend all of our time trying to figure out what the hint is. Either way, it's just enough of a hint that works well for me. We might even erase the hint I chose because it wasn't working well and choose a better one. But we create these hint cards. And now, I want kids practicing with those hint cards. But they're practicing using reasoning and relationships. We want the facts at their fingertips, but we want more than that. We want those relationships also. But, Kim, I got to tell you. I just saw the other day the completely created array cards. So, we've created a set of array cards, ya'll, that are so cool. Let me describe what these look like. So, we had the hint cards I talked about. But the array cards look like closed arrays. So, if I was looking at, say, the fact 4 times 8, it would have a 4 rows by 8 columns. And I'm looking at a grid. And in the middle of it, it would say 4 times 8, 8 by 4. And on that, I see a grid. 4 by 8. Then on the back of that card, it says... Ready, everybody? 32. And the back is blank. There's no grid lines. So, the back is an open array, but it is spatially correct. It's it's a 4 by 8. On the front is a closed array that says 4 by 8, 8 by 4. So, if we're playing with those, we can play several different versions of those. And sometimes we'll do a whole episode on those cards. But I can literally look at that 4 by 8 on the front, and I can ask myself what's the area? Flip it over. Sure enough, it's 32. When I get a bit more advanced, I can play only looking at the back side. So, now I'm only looking at open arrays that simply have the area on it. So, Kim, if you're looking at an array card right now. No grid lines. It's the back side. It only has the area listed, and it says 36. What are you looking at on the card to help you know what? 

Kim  26:26

Yeah.

Pam  26:26

What is it? What's the card?

Kim  26:28

I mean, I think I'm looking at the shape to start with. Like, is it compact? Is it square? Then it might be the 6 by 6. Is it long and skinny? Like, really long and skinny. Then it's a 1 by 36. Or maybe a 2 by 18. But the shape, the dimensions of it, is going to help me figure out what the dimensions are. 

Pam  26:48

Yeah, nice. What if it's not a square.

Kim  26:51

Okay.

Pam  26:52

But it's kind of close to a square

Kim  26:56

Mmm, then it's probably going to be the 4 by 9. 

Pam  26:58

Nice. See! There's like lots of... Look at all the reasoning that can happen. This is a brilliant set of cards that we can use with students that's kind of self differentiating. 

Kim  27:07

Yeah. 

Pam  27:07

Again, we'll do a whole episode on that later. 

Kim  27:09

Yeah.

Pam  27:09

So, Kim, this sequence that we've just talked about, you could do these kinds of messy tasks, these kinds of building foundations for strategy where you analyze strategy tasks. You could do these kinds of practice things with your kids. And you could sequence those with Problem Strings to really help kids build single-digit multiplication and division relationships with students. 

Kim  27:33

Yeah. I'm going to go back to the list that I described at the beginning that if you are a very traditional teacher, then your foundations are memorized facts. But as you were describing, I was thinking about kind of the bigger mathematics that's happening here. We talked about doubling with different numbers. We talked about 10 and 5 and 9. We talked about partials.

Pam  27:55

Partial products, mmhm. Or quotients. 

Kim  27:57

(unclear) property. We built models. So, not only are those essential for facts, but they're also giving you a very firm foundation for what will be to come for these students outside of facts.

Pam  28:11

Mmm, nicely. In so many different ways. 

Kim  28:14

Yeah.

Pam  28:14

Multiplicative reasoning, area, geometry.

Kim  28:17

Mmhm.

Pam  28:17

Like, lots and lots of brilliant. Properties with community property. Yeah. And distributive property.

Kim  28:23

Yeah.

Pam  28:24

With the partial products and partial quotients. Yeah, nicely done. And, ya'll, if you don't want to create this sequence on your own, but you'd like to check out one that we've created, check out our Foundations for Strategies product with Hand2mind. That is the single-digit multiplication and division product that has tasks like this all sequenced nicely for you in an easy to read format. We're super proud of that. If you don't want to create your own, you want to save a little time, then check out our Foundations for Strategies. 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 mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!