Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:09  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is FigureOutAble, we are on a mission to improve math teaching.

Pam  0:16  
Y'all, algorithms are super cool achievements, but they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop and use.

Kim  0:29  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships,

Pam  0:35  
We invite you to join us to make math more figure-out-able. Ba dum dum.

Kim  0:41  
Hi, there.

Pam  0:41  
Hey, there. It's a day. It is. Should we record a podcast? Let's record a podcast.

Kim  0:46  
We should do a podcast. And we should do some math.

Pam  0:49  
Ah, let's do some math. It's been a minute since we've done some. No? Have we done? I don't even know. Let's do some math.

Kim  0:55  
Okay. Alright, so we're going to do a Problem String today.

Pam  0:59  
Mmhm.

Kim  0:59  
And sometimes we get asked questions about Problem Strings and what they should look like at different grade levels.

Pam  1:08  
Mmhm.

Kim  1:09  
Particularly when you do a Problem String with adults, and they ask questions like what would that look like in my classroom? Like, that doesn't feel like...

Pam  1:18  
Especially when... sorry.

Kim  1:19  
Yeah, go ahead.

Pam  1:20  
Especially when it's a large grade range.

Kim  1:22  
Yes. 

Pam  1:22  
So, if I'm like working with fourth grade teachers, I'm probably going to do... Well, I might do an adult-ish string, and then talk about what it looks like in a fourth grade classroom. 

Kim  1:32  
Yeah, absolutely. 

Pam  1:32  
But if I got K-5 teachers, K-6, K-12 teachers in the room, that's a more detailed conversation. 

Kim  1:40  
Yeah.

Pam  1:41  
Should we have that conversation today? 

Kim  1:42  
We should indeed.

Pam  1:43  
Bam. Okay. So, we're only going to do like a specific Problem String, but I think there's some general things that we can kind of take away from if you've done this adult-ish Problem String, what could it look like at different grade levels? So, everybody, we're going to start with an adult-ish Problem String, and then we'll kind of parse it out from there. So, Kim, first question. Again, I'm going to do this with adults. We'll talk later about who else I would do it with. But say I've got adults in the room, I might say, "Alright, y'all, first question we're going to do, not really hard." Oh, I wouldn't say not really hard. What would I say? "First problem. Here we go. 16 minus 10. Alright, nobody's really thinking very hard on that one. 16 minus 10 is everybody?" People will just call out, but go ahead and call out, Kim 16 minus 10.

Kim  2:18  
6.

Pam  2:18  
6. Then, I'm going to say something like, "Hey, nobody needed to really draw anything. But mathematicians have mental maps in their heads, so I'm just going to go ahead and put a 16 on a number line, subtract 10 and land on 6. It's not a very long jump. It's kind of there. We'll just kind of have it there. Mathematicians often have mental maps. Your mental map could have looked like this." Notice that I'm very careful to not say, "Should have looked like this. Must look like this". Like, your mental map could have looked something like this. I'm also not saying thou shalt draw a number line right now in your notebooks. Blah, blah, blah. Okay, next problem. 16 minus 9.

Kim  2:57  
7.

Pam  2:58  
Thank you. 7. I might say, "A lot of you guys are smiling. Did anybody use... Maybe some of you just know that one." But, Kim, did you use the problem before it? Maybe you just know that one. But could you use the problem before to solve
that one?

Kim  3:09  
Sure. Yep. So, if I subtracted 10 before and got 6, now I'm subtracting only 9, so the answer is going to be 1 more.

Pam  3:17  
1 more. And so, then I would draw the 16 minus 10 again, landing on 6. And then you're like, "But it has to be 1 more." So, then I make a little jump up of 1 landing on that 7. Cool. Next problem. How about 37 minus 20. Again, not going to wait very long on that one. Kim, what did you get? 

Kim  3:43  
17.

Pam  3:45  
17. So, I'm going to draw a number line a little shifted to the right. I'm going to do 37, and then I'm going to say out loud, "Bigger jump of 20." Try to make it about twice as big as that smaller jump of 10. And you said land on 17. Cool. Next problem. How about 37 minus 19? "A lot of you guys are smiling." Okay, Kim, did you use the problem before to help you? Could you talk about that? 

Kim  3:55  
Sure. So, if you subtracted 20 before, that got you 17. Now, you're subtracting a little bit less, 1 less, so the answer is 1 more.

Pam  4:04  
And I might at some point do a little bit of talking about, "Wait, we're subtracting, so why are you going the other way?" And especially if a lot of people are like, "Wait, shouldn't you go..."  I might say, "Did anybody get another answer?" And if somebody says 16, then I might be, you know like, "Well, I subtracted that nice number, but then we were subtracting. I kept going." So, I might have more conversation there. For today, we're just going to be like, "Yep, we subtracted." You said it so nicely, Kim. We subtracted less, so we have to adjust up. Okay, cool. Next problem. What if I had 462 subtract 100?

Kim  4:36  
Mmhm.

Pam  4:36  
Okay, go ahead. 

Kim  4:37  
362.

Pam  4:38  
And again, on that one, I'm not going to wait very long. I'm going to go over to the right. I'm going to do 462. And I'm going to do a bigger jump than I've done before, but I really don't have enough room to, so I might do a dot dot dot in the middle of that jump because I can't really do. Not on my paper. I won't have enough room. Maybe on the board I would have enough room. But, you know, much bigger jump. Minus 100. You said that was 3. I'm going to land on 362. Next problem. And now, at this point, with adults, I might say, "Anybody want to guess the next problem that we're..." And I might wait and kind of see what they're... But then I don't. I never let them say it. I might just say, "Want to guess?" But I don't let them say it. And then I go, "It actually is 462 minus 95." And I'm hoping people have sort of... I asked that "guess" question because anybody who's ready at this point gets a chance to look back at the pattern and say, "Okay, it was minus nine. It was minus 19. Ha, I bet this is minus 99." And then I get to go, "95." And it's just enough of a little bit of a ping, a little bit of a zing maybe where they get to go, "Oh, okay." Gives me a chance to kind of smile at them and it doesn't feel too rote. Doesn't feel too mimic everything I'm doing kind of thing. Alright. Kim, so all that talking. What's 462 minus 95?

Kim  5:55  
This time, I can subtract 100 and then add 5 back. So, subtract 100 is 362. And then plus 5 more is 367.

Pam  6:05  
Nice. At this point, I'll make a choice. I could still do another number line, do the first problem and then add on. Or, I might, at this point, just use the 462 minus 100 and just tack on that plus 5 because really, you kind of see both problems in that one number line. It depends on how experienced the people are I'm working with, not really what grade level they teach. It's more if they've been thinking about numbers and using relationships, I might now just use the one number line or redraw the number line again. Does that make sense? I feel like I'm... Okay, cool. So, you said it was 367. I hadn't written that down, so now I'm writing it down. Okay, next problem. Now, again, depending on the experience level, I might at this point say, "Everybody look at the problems that we have so far. I kind of gave you a helper, 16 minus 10 to do 16 minus 9. I kind of gave you a helper, 37 minus 20 that many of you used to do 37 minus 19. I gave you a helper." And I'm kind of putting arrows by these. I'm kind of pointing to them. "Gave you 462 minus 100 that many of you used to help you think about 462 minus 95. I wonder if you could use the patterns that you're seeing. What if I don't give you the helper? Could you create your own helper? What if the problem was 6,371 minus 1,988. I wonder if you could use the same kind of pattern that we've been using. What would the helper be?" Now, I'm going to give people time. I might even have them turn and talk. Kim, if you were going to use the same sort of pattern, what might be the helper for 6,371 minus 1,988.

Kim  7:45  
I would prefer to subtract 2,000.

Pam  7:48  
Ah, look, you didn't even say the 6,371. Now, I might go find somebody that has that. Pretty much, if I've asked for that, I'm pretty sure I'm going to have people that have found that. Depending on the experience level, I might just give that helper problem. I might just say, "Here's the next problem. 6,371 minus 2,000." And then I would give the next one. Kim, how could you solve that second problem? 6,371 minus 1,988.

Kim  8:14  
Yeah, I'm going to subtract 2,000 to get 4,371. But I know that 1,988 is 12 less, so I only want to subtract 12 less than I had than I just subtracted.

Pam  8:29  
Mmhm.

Kim  8:29  
So, I need the other 12 back on. So, if I was at 4,371 when I subtracted 2,000, I'm going to add 12 back, and that is 4,383. 

Pam  8:38  
Nice. At that point, I might have kids turn and talk. How did you create the helper? How does that relate to this problem? Why the 12? Bam. At this point, with adults, I'm pretty... Well, you know, it depends. If we're having pretty good success at this point, I probably won't give the helper again. 

Kim  9:00  
Yeah.

Pam  9:01  
If we're not, people are really struggling, I'm probably giving the helper. For the podcast, I'm not going to give the helper. So, looking back, what kind of helper could you write for 5.2 or 5 and 2/10 subtract 1.9 or 1 and 9/10?

Kim  9:15  
Mmhm. I'm going to prefer to subtract 2.

Pam  9:18  
Alright, so 5.2 minus 2. Which is?

Kim  9:21  
3.2. 

Pam  9:21  
Okay, and how does that help you?

Kim  9:24  
Yeah, I subtracted too much by a tenth, so I'm going to add a tenth back on and get 3.3.

Pam  9:29  
And again, I'm drawing the number lines. We're shifting. Not shifting. We're adding up to make up the difference. Cool. "So, it looks like you..." And I want to start putting generalization words to it. "It looks like you can subtract a bit too much and add back in the extra because you subtracted too much with whole numbers, with thousands, and with decimals. Check it out. But, I mean, you know, maybe only tenths. I don't know if you can do with hundredths. Like, what if I gave you something like 2 and 63/100 minus 97/100 or 2.63 minus 0.97.

Kim  10:02  
It's like 97 cents, so I'd rather subtract a buck. 

Pam  10:06  
Okay.

Kim  10:07  
To get $1.63, 1 and 63/100. And then I subtracted $0.03 too much, so I'm going to add that back on and get 1 and 66/100.

Pam  10:16  
Nice. Cool. But we can't do that with fractions. That would be crazy. I mean, let's try it. What if we had the problem 19 and 1/3. Actually, let me just say, since I'm kind of talking about how I would do this. Fractions is the one that I might just give the helper problem, no matter how well it's going. Especially if I'm working with elementary teachers. It's not a dog on elementary teachers. They just deal with fractions as locations on number lines a little bit less. So, let's go ahead and do the helper problem on the podcast. So, 19 and 1/3 minus 8.

Kim  10:50  
Okay.

Pam  10:50  
What do you got? 

Kim  10:51  
That's 11 and a 1/3.

Pam  10:52  
Alright, so I'm drawing a number line. I'm putting 19 and 1/3 in a location. I'm subtracting 8. It's 11 and 1/3. We now have that landing space. I'm aware, Kim. That please no one look at my iPad right now because that jump of 8 is as big as the jump of 100.

Kim  11:09  
Nice. 

Pam  11:10  
So, I might. You know, I would want to maybe say something to kids of like, "Obviously, these lengths are not all proportional." Okay, next problem. I'm definitely going to ask at this point, "Anybody want to guess the next problem?" Kim, do you want to guess? Do you want to guess out loud?

Kim  11:25  
Well, you have thirds in your previous problem, 19 and a 1/3. So, I'm going to guess that you are doing a third less or two-thirds less than 8.

Pam  11:36  
Yep. It is 19 and a 1/3 minus 7 and 2/3. 

Kim  11:39  
Okay, so since I'm subtracting. When I did 8, subtract 8, it was 11 and a 1/3. Now, we're subtracting a 1/3 less, then I'm going to add 1/3 back. So, that's 11 and 2/3.

Pam  11:54  
So, you just did subtracting of mixed numbers with kind of some craziness, where some people would say you had to borrow. And you're like, "Nah, I'm just going to subtract 8 and then add back in that third." And you ended up having to do 11 and a 1/3 plus a third. Not shabby. Cool. So, we would end that Problem String with having kids talk about, you know, what were we doing every time? Why these? Why in all these number lines do we have this big jump and then this little jump back. And this big jump, and this little jump back. And, you know, like get some words going around that. And we're developing the subtraction Over strategy. Subtract a little over, add back in the too much that you subtracted. Cool.

Kim  12:33  
Which is a great strategy to develop when you have a wide grade range because it's a strategy that applies pretty much at every single grade level, so they can find themselves in it. So, when you do adult-ish strings, they often start with some small number. We did 16 minus 10, 16 minus 9. And then they range, often, all the way up to decimals and fractions. So, let's talk about why you do that with adults. One of the things is that what I've noticed is that when you start a Problem String, you start with something small, and like the tension in the room goes down. Like, people just like relax into it. Like, "Oh, okay." And so, they get a chance to feel the strategy with smaller numbers.

Pam  13:14  
Mmhm.

Kim  13:15  
You're not going new strategy you've never seen before, really big numbers.

Pam  13:20  
Big, crazy or complicated numbers, yeah. 

Kim  13:22  
Yeah.

Pam  13:22  
Mmhm, mmhm, 

Kim  13:23  
Yeah. Also, I kind of just alluded to that they can see how the strategy applies in a variety of grades. So, it kind of gives some weight to a strategy to say like hang on a second. We can do this work in like first grade, and in fifth grade, and in high school. So, that, you know, it demonstrates what a great strategy that it is.

Pam  13:44  
Mmhm.

Kim  13:45  
And a thing to note about your adult-ish Problem Strings is that we just did a lot of problems.

Pam  13:50  
Yeah, it was kind of long.

Kim  13:52  
There are quite a few problems. But that's specifically because you are with adults.

Pam  13:57  
14. I just counted. 14 problems.

Kim  14:00  
Yeah. Yeah, that's not a recommendation to do that particular string in its entirety in a classroom. It's you kind of have a different point, I think, when you're doing these strings with adults in a workshop. Yeah, so you can get a lot more in faster because maybe you're working with adults. But you also want them to see the stretch that can happen.

Pam  14:22  
Mmhm. Yeah, nice. 

Kim  14:23  
So, sometimes people ask about strings for their particular grade level. Or in a workshop, they would say, "What of this would you do with fourth grade kids? What of this would you do with like second graders? Like, this string wouldn't be good for second graders. It has decimals and fractions." So, we thought we would talk about this particular operation, subtraction, and this particular strategy, the Over strategy, and name out loud what we might do for different grade levels. So, when you're sharing, let's talk about which problems we would keep and why. And then maybe what we would add in for a particular grade level.

Pam  14:59  
Yeah, love it. So, let's say that. Let's say. I'll just say. That we believe this is a good strategy for first grade and up. 

Kim  15:08  
Yeah.

Pam  15:10  
Like, intentionally teaching it. Now, maybe you got some kids in kindergarten that are ready for it, but first grade. So, I would keep the first problem in first grade. 16 minus 10, 16 minus 9. I said problem. It's really the pair of problems. But then I would stay in the teens. So, the second problem might be 13 minus 10 minus 9. And this next set might be 17 minus 10 minus 9. And then I might end with 15 minus 10 minus 8.

Kim  15:35  
Yeah.

Pam  15:36  
Just for that little bit of at the end where it's not too formulaic, but you got kids really thinking. And, in a big way, it's not about getting kids, "Oh, I am learning a strategy. It is..." First grade kids will be thinking about what is it from a teen minus a 10? Teen minus a 10? Oh, that's that number hanging around, right? 17 minus 10 is 7. 16 minus 10 is 6. Like, that's part of the meaning of teens. So, we're building that. We're building a sense of place value a little bit in the teens. And we're helping kids think about the Over strategy. What happens if you back up too much? So, yeah. That's what it could look like in first grade.

Kim  16:12  
Can I interrupt you to say that like when you do that teen and then you subtract the 9, the problems that you just named that were the yuckier problems are yucky. Like, they are some key ones.

Pam  16:24  
They're often missed.

Kim  16:25  
Yeah! Yeah, and so first graders getting to experience that. It's pretty great.

Pam  16:29  
It's nice, yeah. And, in a Problem String, they're getting experience where they get their answer confirmed, checked, often. You know, you could go send kids off to do that many problems, and they can do them all wrong and not get good feedback. 

Kim  16:52  
Yeah.

Pam  16:52  
But they're getting feedback because you're doing them all together. Okay. Second grade. I probably keep the first two pairs of problems. So, 16 minus 10 minus 9, 37 minus 20 minus 19. But then I'm going to stay in the two digits, sort of under 100. So, then I might do 52 minus 30. 52 minus 28. Here, I'm going to change it up. It's not 29, 28. Then I might do 83 minus 50. Notice, that's a big jump. 83 minus 50. That's a big ol' They're thinking about that. And then 83 minus 47. So. Yeah. Tweaking that last problem again. But also, it's not just... If you look over the whole string, it was minus 10, then minus 20, then minus 30, then minus 50. The next set, the next Problem String I do, I might do a minus 40, minus 20, minus 30, minus 60, but kind of staying in the 100. 

Kim  17:39  
Yeah. Yeah, nice. So, in third grade, I might do one pair that's like 37 minus 20, 37 minus 19 that we had in there, and then jump to the 462 minus 100 minus 95.

Pam  17:49  
Mmhm.

Kim  17:49  
Those are nice starter. But, like you, I'm going to hang longer in the like hundreds. So, maybe 546 minus 200, and then minus 198. And then like you, the jumping up from the big jump. A next nice pair might be 813 minus 500 and minus 497. So, you're spending a lot of time in the hundreds. Which is, you know, a lot of the bulk of third grade.

Pam  18:14  
Mmhm.

Kim  18:14  
But then we go back. Then we go back to one of the problems, which was the thousands. So, 6,371 minus 2,000 and minus 1,998. So, the first problem is like a little entry and the last problem is a stretch, but the majority of it is in the hundreds. 

Pam  18:29  
Nice. 

Kim  18:30  
We talked about this, and we agreed that for fourth and up, we would just pick all the problems from the middle for this particular string. So, starting with the 462 minus 100 minus 95. The 6,371 minus 2,000 and minus 1988 into those two pairs of decimals. 5.2 minus 3 and minus 1.9. 2.63 minus 1, minus $0.97, and then into that stretch of a fraction problem. You know, we just cut off a little at the top and at the bottom. 

Pam  19:04  
Mmhm.

Kim  19:04  
And then with adults, you mentioned the problems that you would do. We did the adult-ish string. But you left off some of the helpers when you were talking about it.

Pam  19:14  
Yeah.

Kim  19:14  
So, again, same thing. Do the stretch of the problems with adults, but leave off the helpers unless they're necessary. Like, as you're facilitating and you see the need for them, you can put them back in. But the aim would be not to.

Pam  19:27  
And even though you just said on the last one adults, I think that could... The fourth and up and the adults. Really, it's pretty much the string that we did here on the podcast, where you're then making judgment calls about...

Kim  19:39  
Sure.

Pam  19:40  
...when do you ask for a helper problem? When do you say, "Anybody guess the next problem?" You know, you're trying to keep everybody kind of challenged enough and everybody secure enough in what's kind of going on. 

Kim  19:52  
Yeah. 

Pam  19:53  
And then you're taking note. You know, how did kids deal with the decimals? How did they deal with the fractions? Are they able to think about 2,000 or some numbers in the thousands minus 1,000. Or, you know like. And they're practicing rounding, and estimation, and reasonableness. And they're getting a sense of what it means to subtract a bit too much, and then add it back in. And if you're in middle school, you could do some work with subtracting. When you do that, you're sort of... Like, let me use one of the problems, 462 minus... Actually, let me use an easier one. 37 minus 20. And then you were when you did 37 minus 19, you're kind of doing 37 minus 20 minus 1. So, if I do 37 minus the quantity 20 minus 1. And then I distribute that negative, you end up with 37 minus 20 plus 1. 

Kim  20:43  
Yeah. 

Pam  20:43  
So, there's some nice work that you can do making sense of distributing that negative. And, you know, sort of an algebraic feel for what's happening. So, lots of good work can kind of happen at the upper grades, while you're helping kids build the numeracy that maybe they didn't have a chance to build younger.

Kim  20:56  
Yeah.

Pam  20:57  
Yeah, nice. Hey, y'all, we're going to write this up. We've been asked so much for this kind of thing that we're going to write this up and put it  in our blog. So, mathisfigureable.com. I think it's /blog. But if you just Google mathisfigureable.com or Math is FigureOutAble blog, you'll get there. And then just check out the blog that we're putting out right about the time that this podcast comes out. And you'll see. I know we went through these numbers kind of fast, but we'll kind of write it up, so you can kind of get a feel for what it would look like to take an adult-ish string and kind of make it fit for different grade levels and why. 

Kim  21:29  
Yeah. 

Pam  21:29  
Alright, y'all, thanks for tuning in and teaching more and more real math. To find out more about the Math is FigureOutAble movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!