# Math is Figure-Out-Able!

Math teacher educator Pam Harris and her cohost Kim Montague answer the question: If not algorithms, then what? Join them for ~15-30 minutes every Tuesday as they cast their vision for mathematics education and give actionable items to help teachers teach math that is Figure-Out-Able. See www.MathisFigureOutAble.com for more great resources!

## Math is Figure-Out-Able!

# Ep 225: Place Value Tasks

Do your students struggle with place value? In this episode Pam and Kim discuss great tasks to build real place value understanding for students of all ages.

Talking Points:

- "Adding a zero" vs shifting place value
- Place Value versus Place Labeling
- Avoiding rules that expire
- Build a Number activity
- Place Value Mini-Workshop!

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**Pam **00:01

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

**Kim **00:10

And I'm Kim Montague, 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 **00:18

We know that algorithms are amazing human 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 **00:32

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

**Pam **00:38

We invite you to join us to make math more figure-out-able. I've been thinking a lot about relationships lately. Hey, and maybe that, maybe that is because I'm doing a ton of what we call Spark sessions. So, we call them Spark sessions because they're virtual, 60 minute sessions, where I get on with schools around the world. And the intent of that session is to spark people's interest in the Math is Figure-Out-Able movement and what it means to teach math that's figure-out-able. So, we call them Spark sessions. And, Kim, I was going to tell you one of the things that keeps happening, you know me. Haha, I know you know me. You're the one who's always telling me not to do this. And you've actually been super helpful because I will tend to... It's almost like a stream of consciousness. Like, I start something and whatever occurs to me, I just kind of go there.

**Kim **01:27

I know.

**Pam **01:30

You're clear on that. The first few times that we did workshops together, you were like, "What are you doing?" And I was like, "What am I doing? I always do this." Yeah, yeah. I'm like, "I always do this." You're like, "But that's... Like, what? Like, stop doing that." So, I have definitely gotten better at it. And let me tell you a place I've gotten better at it. So, often, not not all the time, but often we will do... Well, I'll say always. We always do a mathematical experience, so people can feel what it looks like and feels like to math the way mathy people math and how to teach it that way. And one of those mathy experiences is often where I will do something with 10 times 27 is one of the problems. Or 10 times something. 10 times 17. 10... Usually, it has a 7 in it just because sevens are often those goofy numbers. And so, I'll say, "You know, what if we had 10 packs of gum that have 27 sticks in it." And people smile, and they're like, "Well, 10 times 27 that's duh. That's just 270." And in that moment, there is often probably 50% of time someone will say, "I just added a 0." And, Kim, here is where I want you to just be so proud of my self control. Because what I don't do in that moment is go off on the long diatribe that I used to about place value. Like, I used to go, "Oh, add a 0. Let's chat about that over here." And then I would... I'd literally I would walk over to a different side of the board or on my iPad I would scroll up, and then I would say, "Alright, lesson on place value. Here we go." And I would do this bit on place value. And 10 minutes later, I'd go back, and I go, "Alright, where were we? Alright, 10 packs of gum. 270. Now, that you understand everything you need to know about place value..." and we sort of move on. So, now I tend to say something like, "Ah. So, you, if you scale the packs times 10, then you can scale the sticks times 10. And there's this nice thing in our base 10 number system that when you multiply by 10, there's this 0 thing. Maybe we'll get a chance to talk about that later." And I just... I have to say those words. "Maybe we'll get a chance to talk about this later," so that I can move on and do the rest of the string, even though I never do have a chance in that 60 minutes to come back to it.

**Kim **03:32

Yeah.

**Pam **03:33

I just have to let my brain think that there's a possibility that I'll be able to come back to it, and then I'll be able to come back to it.

**Kim **03:40

Yeah. That's... Well, that's... The danger of that, you know, is you can't let it go completely. You want to address, but you can't address it fully because you're only going to see these people for 60 minutes. But, you know, in a classroom, you could address that in a different time, right? You're going to see those kids again. And so...

**Pam **03:58

Right, right.

**Kim **03:59

Yeah, I love that you you stay focused on the goal of a string.

**Pam **04:03

Thank you. And if I was in my classroom when I teach at the university, I would literally go over to the side of the board, and I would write down. And I'll do this in a long workshop as well. I'll go over and kind of in a back burner space, I'll write "add a 0" or "place value" or something that will, you know, sort of spark this moment that happened. We're coming back to it.

**Kim **04:22

Sure.

**Pam **04:23

Yeah, I don't, I don't usually have time do that in an hour, but. But for two reasons, A, that's good, because we need to come back to it. It's an important mathematical thing. And B, it allows me to move on.

**Kim **04:33

Yeah.

**Pam **04:34

If I'm doing that, yeah.

**Kim **04:35

Yeah.

**Pam **04:36

Cool. Okay, so, Kim, why... People are listening to the podcast and like, "Pam, why are you getting your underwear in a wad? Like, what? What? Why do you care that people are saying, you know, I just add a 0." Yeah. So (unclear).

**Kim **04:49

Well.

**Pam **04:50

Yeah. Why do we care?

**Kim **04:51

Well, first of all, you're not adding a 0. It's...

**Pam **04:53

I mean.

**Kim **04:54

...times 10 and you're... So, then teachers say, "Well, I don't. I tell them not to say that. I tell them place a 0." And, you know, I get it. It's not about...

**Pam **05:04

Put a 0?

**Kim **05:05

...just replacing the words. It's about helping students understand what's actually occurring.

**Pam **05:11

So, it's not okay to just change the words.

**Kim **05:15

Put a 0 at the end.

**Pam **05:16

Put a 0 at the end. Place a 0. That's that's not enough to do that.

**Kim **05:19

I mean, we really want kids to understand that what's happening is they're scaling by 10, that it's 10 times bigger. But then also another issue is that what happens when you multiply a decimal by 10? Like, we cannot be saying things early on for rules that expire later.

**Pam **05:37

Yeah, that's a rule that expires because if I have something like 3.2 times 10.

**Kim **05:42

Yeah.

**Pam **05:42

And if I just add a 0, is 3.2 times 10, 3.20?

**Kim **05:47

Yeah.

**Pam **05:48

Like, no. So, that's definitely a rule that expires. So, that's a reason not to to say it. You know, it's mathematically incorrect. It's a rule that expires. But also, there is an opportunity here that we have not known. I think I didn't know how to take advantage of to help students with place value, to help them actually gain a better sense of of place value. Yeah.

**Kim **06:14

And we talk about that a lot in in episode 100. We did a whole episode on place value versus place labeling, (unclear).

**Pam **06:22

That was kind of a fun episode, right?

**Kim **06:24

Yeah.

**Pam **06:24

Because it was 100

**Kim **06:25

Yeah. Yeah, yeah.

**Pam **06:26

Yeah

**Kim **06:26

Gosh, it feels so long ago. And in that episode, we talked about place value versus labeling. Which is typically what happens in early classrooms. We do a lot of talk about what digit and what place is it in, but really not developing with students what the value of numbers are. And so, we did that episode...

**Pam **06:43

And, yeah, let me interrupt you. And just to give teachers their their due, it's what textbooks have done.

**Kim **06:49

Oh, yeah.

**Pam **06:50

Textbooks, for the most part, if you look at anything in a textbooks textbook today that says place value, it's mostly, if not all, place labeling. It's all about labeling places. It's not really about the value of the place of the digit at all. It's just labeling. And kids could actually be fairly mindless on most of that place labeling stuff. So, yeah, episode 100 you could totally check out if you'd like to know more about place value versus place labeling. I'm sorry, I interrupted you.

**Kim **07:19

Yeah, no that's okay. In that episode, we talked about some of favorite routines and activities that we do to develop place value. So, that's a great one to go to. But also, we wanted to share a little bit more today about a few new things that we've been talking about or for a few different things in that. In that product that we just talked about last week. And...

**Pam **07:36

Yeah.

**Kim **07:37

The Hand to Mind lessons. We wrote about build a number. And so, in build a number, it's really about deconstructing a number and renaming numbers in different ways that still maintain the same value. So, we might have said... I don't know. Maybe one of the numbers we used was 78. And we would say something like, "Okay, so we know that it's 70 and 8. That's that's a fairly traditional way of describing the value of that number. But what else can we say?" What else can we say about that number? You want to give me one. What's another name, another...

**Pam **08:12

60 plus 18.

**Kim **08:15

It also has the same value of 78. What else?

**Pam **08:19

How about 30 plus 48

**Kim **08:23

Nice.

**Pam **08:24

Or I was wondering if I was going to flip that or not. So, like 40 plus 38. What I was actually saying in my head is why didn't I do 40 plus 38 before I did 30 plus 48. Which I don't actually know the answer to that.

**Kim **08:37

Yeah,

**Pam **08:38

Which is why I paused. (unclear).

**Kim **08:39

So, one of the goals of that routine is that we're really having kids understand what's happening. And it leads to things like if you have 314, it's not just 300 and 10 and 4. There are lots of other ways that we can describe the value of 314, and so...

**Pam **08:40

(unclear). Yeah, you do it. Alright, 314. Go. What's another way?

**Kim **08:45

I'd love to write it down, so I can think. So, that's 31 tens and 4 ones. So, 310 and 4.

**Pam **09:08

Mmm, mmhm.

**Kim **09:09

There's also within that, 200 and 114. Makes up 314.

**Pam **09:15

Nice. Yeah.

**Kim **09:16

It has 314 ones.

**Pam **09:19

Uh huh. Uh huh. Yeah, yeah. So, if I went kind of big to small, you could say 314 ones, 31 tens with 4 ones left over, or 3 hundreds with 1 ten left over and 4 ones left over.

**Kim **09:34

Yeah.

**Pam **09:35

And notice, notice that leftover idea. I think is kind of important.

**Kim **09:39

Yeah. And you know what? I actually had this conversation with a fifth grade class one time, and a kid said, "Well, I think that's 31.4 tens.

**Pam **09:50

Oh, I love that.

**Kim **09:51

Instead of the leftovers. I was like, "Yeah, you go." Yeah.

**Pam **09:54

Ooh, yeah.

**Kim **09:55

Anyway, so we love build a number as...

**Pam **09:57

Does that mean it's also 3.14 hundreds.

**Kim **10:02

Pi hundreds. What?

**Pam **10:05

That's funny. I didn't even think about. It's approximation for pi, but yeah.

**Kim **10:09

Yeah, yeah.

**Pam **10:10

That's funny. Did you do that on purpose?

**Kim **10:12

I like hundreds of pi's. Okay.

**Pam **10:15

Yes, you do.

**Kim **10:16

Alright, so there's another, another actually a kind of a task that we introduce and then revisited. That's, that's one of my favorite things is when you do a task, and then you layer on over time.

**Pam **10:28

I love it.

**Kim **10:29

(unclear) context that's really exciting, and then you do different things with the context. Anyway, so there was a task that we wrote about, about reading minutes. You know, lots of younger grades, they read as a requirement for school, and so we thought we'd do something with reading minutes. And so, in this in this context, students read, and then they kept track of their reading, and for every 10 minutes they got a point, and for every 100 minutes they got a prize. So, there's lots of nice things that we did with the points and the prizes. And notice that they are 10 and 100. And so, we able to say things like, "How many points give you a prize? How many minutes give you a point, so how many minutes give you a prize?"

**Pam **11:13

So, are you saying that if a class gathered all of their reading minutes and they, over a period of time, had read 462

**Both Pam and Kim **11:21

Minutes.

**Pam **11:22

How many points would they get? Yeah,

**Kim **11:26

And then how many prizes would they get?

**Pam **11:28

So... Yeah, go ahead. Go ahead. 462. How many points would they get?

**Kim **11:32

Well, do you get a partial point? I don't actually know what we said about that. But they will get 46 points.

**Pam **11:38

Okay, okay.

**Kim **11:40

And 4 prizes.

**Pam **11:41

Why?

**Kim **11:43

Because there's 4 hundreds for the prizes, and you get 100 point... 100 minutes per (unclear)

**Pam **11:52

100 minutes gets you a

**Both Pam and Kim **11:53

Prize.

**Pam **11:54

Okay. And since there are 462, they they haven't made it up to the 500 yet, so they don't get 5 prizes, but they can get 4 prizes. (unclear).

**Kim **12:02

But we also got to write about I Have, You Need, and then we were able to say, "And how many minutes do they need to get to the next point, and the next prize."

**Both Pam and Kim **12:20

Yeah.

**Pam **12:20

Yeah. So, for 462, they would need 8 more minutes to get to the next point.

**Kim **12:24

Point, mmhm.

**Pam **12:24

And they would need 38 minutes to get to the next prize.

**Kim **12:25

Mmhm.

**Pam **12:25

Yeah, nice, nice. So, I like the, I like the prize. I like the prize for reading minutes. And and Foundations for Strategies, has both of those tasks in it, this build a number that we just described and these prizes for reading minutes.

**Kim **12:40

Yeah.

**Pam **12:41

Super cool. Worked in. Like, right? It's kind of like we do it a little bit and a little bit more and kind of these different. Like, you said, bringing in I Have, You Need. And, yeah. The sequencing. We're all about sequencing. So, super, super, super cool.

**Kim **12:53

Yeah, so one of my favorite things that I get to do is write some things for... I don't even know if you know that I love writing things for Journey. And so, one of the things that we do there in our support, implementation support group, is we provide things that we think would be (unclear) a classroom. And so, we just recently got to put in some place value mini sheets. And so, I'm going to actually give you some problems. So, on those sheets, you get to explore relationships between numbers. And this one was specifically geared around scaling and thinking about place value. So, I thought I'd give you some problems. You always give me problems, and I have to think on the fly.

**Pam **13:32

You're going to make me think on the fly.

**Kim **13:33

Yeah, I am going to have you do that. And in this particular activity, there's a given. And so, there's... What's really nice is there's access for kids who even still need to do some more work with their facts. But here's your given. So, the given is 3 times 7 is 21. You could write that down if you wanted.

**Pam **13:56

I just did with my pen. (unclear).

**Kim **13:59

Mmhm. That's wrong right off the bat. So, okay, given that 3 times 7 is 21, notice 30 times 7.

**Pam **14:09

So, I've just written 30 times 7 below it, and I'm looking at the relationship between 3 times 7 and 30 times 7 and 30 is 10 times 3. So, the product is also going to be 10 times 21, which is 210.

**Kim **14:24

Okay, (unclear).

**Pam **14:25

Scalable by 10.

**Kim **14:26

Yeah, great. The next one is 30 times 70.

**Pam **14:31

Cool. So, now, from the 30 times 7, now I've scaled the 7 times 10. So, that product, the 210, is also going to scale times 10. And 210 times 10 is 2,100.

**Kim **14:44

Okay, (unclear).

**Pam **14:45

So, I've just actually, I've actually just written... Is that what you want me to say is what I've written? No.

**Kim **14:50

The problem?

**Pam **14:52

Well, you were just going to say, "Can you..." and I interrupted,

**Kim **14:53

Well, I was going to say can you relate it to the first given?

**Pam **14:58

Ahhh. Yeah. So, the first given was 3 times 7 equals 21. So, to the 30 times 70, the 3... The 30 is 10 times the 3, and the 70 is 10 times the 7, so the 2,100 has to be times 10, times 10 or times 100 times the 21.

**Kim **15:18

And how nice would it be for kids to be able to recognize that's 3 times 7 times 10, times 10, or 3 times 7 times 100. Love that. Okay, nice job. And then the next one is (unclear)...

**Pam **15:29

Ooh, and before you go... Before you go on. Sorry.

**Kim **15:31

Yeah.

**Pam **15:31

And that 21 times 100, we also call that number 2100. Which is why we call that number 2100 because it's 21 scale times 100 Okay, go ahead. What was the next one?

**Kim **15:41

The next one is 3 times 70.

**Pam **15:43

I feel like we had that one before, but we didn't. We had 30 times 7. Now, we have 3 times 70. So, I'm going to actually relate it. I'm going to relate it to the very first one. So, the first one was 3 times 7 is 21. The 70 is 10 times the 7. So, I know the product is going to be 10 times 21, which is 210. But I'm also going to relate it to the 30 times 7. So, 30 times 7 and 3 times 70, I've just re-associated a multiple of 10, a times 10 in there.

**Kim **16:15

Mmhm.

**Pam **16:15

So, if that times 10 is still there, I've just taken it from the 30 and stuck it on to the 7 to make 70. Stuck it on with multiplication. Then I... Then the product remains the same, and it's, and it's 210

**Kim **16:29

Yeah, yeah. And I could see on the board writing 30 times 7 equals 3 times 7 times 10 equals 3 times 70. Okay, last one. You ready?

**Pam **16:42

Yeah, and you could kind of... You could even written one or more of those where the parentheses change. Just to kind of show the re-association. Okay, I'm ready for one more.

**Kim **16:49

Last one is 300 times 7. Tell me all the connections you can find.

**Pam **16:54

Every connection possible.

**Kim **16:56

Okay, maybe just two.

**Pam **16:58

The given was 3 times 7 equals 21.

**Kim **17:01

Yep.

**Pam **17:02

And so, I'm going to go that the 300 is 100 times that. The 300 is 100 times the 3, so the product is going to be 100 times the 21. And 100 times 21 is 2100. If I was to go from one other one, I think I would go just... I don't know why. Today, I would go from the 30 times 70. And I'm going to do that re-association thing. So, instead of 30 times 70, I'm going to take the 10 from the 70 and keep that. And that'll be just 7. And stick that times 10 with the 30. So, now the 30 times 10 becomes the 300 you just gave me, and I'm left with the 7 because I just grabbed the 10 out of it. So, it should have the same product as the 30 times 70. 30 times 70 should have the same product as 300 times 7.

**Kim **17:52

Yep. So, I love this sequence that could be delivered like a Problem String, where you give kids a problem at a time, you ask them to share how they see it. What we chose to do with the place value mini sheets was give all the problems. So, not like a Problem String. We give the problems, and then we ask students to draw the connections they find. And so, if you find one connection, fantastic. If you find more more connections, fantastic. And all the time you're strengthening times 10. What we didn't say here, that is also possible, is divided by 10. So, we could have started with the 300 times 7 and scaled the other direction.

**Pam **18:29

You know, I might, I might have been tempted to start with a 30 times 70. But, but you could probably do both, right? Like, yeah, just depending on... You could do the same. You do one on one day and the other on the other and go both. In and out, in and out.

**Kim **18:42

Yeah, yeah.

**Pam **18:43

And to be clear, it's not just about like add a 0, move a 0, move the decimal point, shift the numbers. It's really the language that we just used, which trying to get out of students. You're trying to... You know like, it's the reasoning place value wise. Yeah.

**Kim **18:58

Yeah.

**Pam **18:58

Nice. So, Kim.

**Kim **19:01

Yeah?

**Pam **19:02

If our listeners are interested to learn more about place value. We created a place value mini workshop, and it's amazing. It's a mini workshop, so it's about two hours of run time. It's got the content and activities to do with your students. It's got from young stuff to do with super young students, all the way through, through upper elementary middle school, through scientific notation. If you want to help your students, you want to help you and your students learn more about place value not just place labeling, a super good place to check that out would be mathisfigureoutable.com/mini. M-I-N-I. (unclear).

**Kim **19:40

You know what, if I were, if I were on a grade level team today, I would go to my principal, and I would say, "You know what? We need one of these mini workshops for all of our teammates, so that we can have a common language. We can do some activities that are meaningful, but we all know about them." It's super reasonably priced. Yeah, you can... Go, hit your boss up.

**Pam **20:02

That's right. That common conversation can mean all the difference for you and your grade level and for then all of your students moving on to the next grade level. So, while you're at it, you might want to get the grade level below you, underneath you, in on it, so that those students come to you with this place value. Place value. Like, we're actually happening in them. Yeah, nicely said. Alright, 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.