May 09, 2023
Pam Harris
Episode 151

Math is Figure-Out-Able with Pam Harris

Ep 151: Subtraction Facts

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Math is Figure-Out-Able with Pam Harris

Ep 151: Subtraction Facts

May 09, 2023
Episode 151

Pam Harris

How can we help students own subtraction facts? In this episode Pam and Kim describe some basic strategies and then walk us through facilitating two Problem Strings to build subtraction relationships.

Talking Points:

- I Have, You Need - see Episode 7 for more on I Have, You Need
- Patterns in a subtraction chart
- Generalizing about subtracting 9
- Noticing differences of 1 or 2
- Diagonal patterns of Constant Difference
- Subtracting to 10 Problem String
- Using Doubles Problem String

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

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How can we help students own subtraction facts? In this episode Pam and Kim describe some basic strategies and then walk us through facilitating two Problem Strings to build subtraction relationships.

Talking Points:

- I Have, You Need - see Episode 7 for more on I Have, You Need
- Patterns in a subtraction chart
- Generalizing about subtracting 9
- Noticing differences of 1 or 2
- Diagonal patterns of Constant Difference
- Subtracting to 10 Problem String
- Using Doubles Problem String

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Pam:

Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim:And I'm Kim.

Pam:And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We believe we can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim:Hey, Pam?

Pam:Hey, Kim?

Kim:It's Challenge Week! I'm so excited!

Pam:We love challenge week!

Kim:Oh, my God. It's one of the most fun weeks of the year. We do it a couple times a year. But we love, love, love seeing your faces, and I get to talk to so many people in the Facebook group. So, I'm super excited about the back half of this week. Not that I don't love the podcast, but you know.

Pam:No, it is so one of the best things that we do all year. It's totally free. We love it. Obviously, if you're listening to this podcast at some other time of the year, just know get on our mailing list and join in those free challenges because they are a blast. Alright.

Kim:Super fun. Alright, down to business. If you haven't been tuning in lately, then you've been missing some great chatter about subtraction and how important it is that we give kids solid experiences with subtraction because it affects so many of the operations, the other operations, and strategies that we love. So, today, we're going to continue from last week, chatting about subtraction facts.

Pam:Yeah, so if you didn't listen to last week, really do because we talked all about memorizing subtraction facts. What are our favorite ways of helping students really own subtraction facts, really getting them into long term memory with a schema, so that they stick better and that it's all more and more Figure-Out-Able? So, Kim, what's one of our favorites?

Kim:Well, you know, I'm going to mention the first one that I love is I Have, You Need. And we talk about it all the time because it's so powerful.

Pam:I Have, You Need. Yep.

Kim:Like, the silliest little routine but super helpful at a young age. We're going to just mention it here because we've talked about it before. We'll link that episode in the show notes.

Pam:Well, and to be clear, you only call it silly because you made it up. So, you're just trying to like, "I'll be, you know, humble and everything."

Kim:Well, I mean, honestly, there's nothing like super magical about it. It's fun mathematically, but the routine itself is basic.

Pam:I mean, I Have, You Need is amazing!

Kim:There's not a million things to make. I mean, I'm in for not a lot of prep. And...

Pam:Yeah, it's good.

Kim:...it's okay to do it in lots of ways, over lots of experiences. Partners, not partners, whole class, different numbers...

Pam:Absolutely.

Kim:...change things up. Yeah, I'm in for that.

Pam:Alright, so if you have not listened to an episode on I Have, You Need. We'll put the episode number in the show notes. But check that out. If that's new to you, if it's the first time you listen to the podcast, I Have, You Need, wonderful classroom routine, all grade levels. It's great.

Kim:Yeah. So, at a young age I Have, You Need with 10. We're talking subtraction facts, so I Have, You Need with 10 would be super helpful. But then, also, once your kids are working and have made sense of the 10 a lot, it would be really helpful for them to work on I Have, You Need with 20. And the idea that within the 20, there are two 10's and one of the 10's is being messed with, with I Have, You Need, but the other 10 is just kind of being left alone. So...

Pam:And I almost want to back you up just a little bit. Let's just do a quick example with a total 10. So, if I say the total is 10. I Have 7, You Need?

Kim:Three.

Pam:Three to make. 10, right? So, that feels like addition. 7 plus 3 is 10. Then, we want to help kids make a connection that therefore, if we own that 7 and 3 are partners of 10, then I now know the answer to 10 minus 7. Three. And I also know the answer to 10 minus 3 is 7. And that takes some time and some development, but that they own that. And so, then, you're saying once we own that, there is that 10 in the 20.

Kim:Inside the 20, mmhmm.

Pam:Yeah, so if I play I Have, You Need total 20. Kim, if I have 17, what do you need?

Kim:Three.

Pam:Three.

Kim:And so, that 10 in your 17. The first 10 is just kind of sitting by itself. But you haven't same 7 and 3 partnership that you had in the 10.

Pam:Yeah. So, if I say I have 12, what do you need?

Kim:Eight.

Pam:Eight. And you're thinking about, "There's that 10 hanging around, but there's a 2..." It's almost like I said, "I have 2, what do you need to make 10?" It's the 2 and the 8 you're kind of using. And, you know, you have to help kids kind of experience that 10. Kind of, sort of ignore that extra 10 and the 20. We're kind of focus on those partners. And then, you could do the same thing with 30. And not necessarily that you would play I Have, You Need but that you make the connection. So, if you're doing 30 minus 8.

Kim:Yep.

Pam:Or 30 (unclear), then can you use the 8 and the 2 in that 10 that's between the 20 and 30. Etc, etc. Yeah. Cool. Cool.

Kim:What's another one?

Pam:Do we want to mention... Sorry. Do we want to mention... My brain's all over the place. That, similarly, if we've done partners of 100 with I Have, You Need. And if I say, I have 63, and you say...

Kim:Yeah, but we're talking about subtraction facts today, Pamela, so I (unclear).

Pam:Oh, don't go higher. That's why, Because I was just going higher.

Kim:What can we do with subtraction facts. (unclear).

Pam:Never mind, stay in subtraction facts. Stay in subtraction facts. Not any subtraction. We're talking about subtraction facts. Okay, cool. Thank you. So, another thing that we can do. There goes my vertical brain. Another thing that we can do is think about groups of problems and what they have in common, right?

Kim:Oh, yeah, yeah.

Pam:What relationships are there among groups of subtraction fact problems? How they're related? How they're connected? So, you have a favorite one. Give us that one.

Kim:So, on a subtraction chart, right? If we're looking at a subtraction, all the subtraction facts, then you know I might put up that chart for five minutes a day for a week, or fifteen minutes at a time, and have kids talk about, "What do you notice? And what patterns do you see?" One of the ones that I think is super fun...

Pam:And hey. Sorry, before you go into the one. Maybe I'll just describe that just a little bit more. So, we might look at a chart where I've got the numbers 0 to 20 across the top, and then 0 to 20 going down the side, and then we're kind of looking at where those coincide. And if we're just dealing with positive numbers, then I would look at where, you know, 20 minus 5 would be in the cell, where the 20 column and the 5 coincide.(unclear). In that cell, we're talking about 20 minus 5. And if we go down to the cell below it, that would be 20 minus 6, and the cell below that would be 20 minus 7, etc.

Kim:Yep.

Pam:So, that's the chart. So, we might pull that chart out, and then we might look for commonalities and groups of related facts.

Kim:Yeah. So, one really cool noticing is that every time you're subtracting 9, you can be thinking about the teen numbers, and you can be thinking about,"It's always going to be one more to get to 10 plus all that extra that was in the teen number. So, if it was like 17 minus 9, then you could think about the 1 that you need to get to 10, plus all the extra 7.

Pam:So, you're really using the difference interpretation of subtraction here.

Kim:Yep.

Pam:You're thinking about if it's 17 minus 9, you're like,"From 9 to 17..." You think about that 1, and then that extra, the kind of the teen-ness of it, right? The 10 to the teen?

Kim:Yeah, and it's true for every 9. Right, every subtract 9...

Pam:Do 13 minus 9.

Kim:So, 13 minus 9 would be... From 9, the 1 to get to 10, plus all that extra 3.

Pam:So, it's always 9 to 10. So, it's always that 1. And then it's just the extra to get to the...

Kim:And it's a really nice transition for kids who are counting by ones to be thinking about, "Oh, I know about teen numbers. It's just all that extra, all those 3." Instead of counting 1, 2, 3 from 11, 12, 13. Yeah. So, getting them used to the idea that you know what that teen number is made of? So, you can think about that whole extra chunk beyond the 10 all in one group.

Pam:And so, this is just screaming at me. This is one more example of, what we're not really suggesting is that you go teach this strategy to kids, right?

Kim:Right (unclear).

Pam:It's not... We're suggesting we notice it because we're in that noticing. We're helping kids go, "Oh, that really is true that from 10 to the 13, it's 3. And from 10 to the 16, it's 6. Or from 10 to the 17, it's 7. And from..." It always is that sort of extra from the 10 to the teen. And then, it would just be that extra 1 because we're subtracting that every time.(unclear). We're always looking at the difference being(unclear). Those generalizations. Building numeracy, building the mathematical relationships that we want kids to own.

Kim:Yeah.

Pam:Because our goal is more than just those subtraction facts, right?

Kim:Mmhmm.

Pam:Yeah, totally cool. So, another group of numbers that we might look at on a subtraction chart would be numbers that are close to each other. So, if I'm going to subtract something like 9 minus 8, or 7 minus 6, or 3 minus 2, that we can think about those, again, with a difference perspective. How far apart are they, if we're looking at subtraction as a difference? Well, if the numbers are always just 1 apart, then they're always just 1 apart. Like, you know, 9 minus 8 is going to be 1. Like, 4 minus 3 is going to be 1 because they're just... All those numbers are 1 apart. What if there were 2 apart? So, if I'm looking at a subtraction fact, and the numbers are 2 apart. Bam! Then, the answer to that subtraction problem is 2. That would be another sort of group of numbers that are related that we kind of look at.

Kim:Yeah. So, I think once we've spent some time working with these strategies, I love the idea of putting up this chart and saying like, "Talk to me. What do you notice?" You know, I'm actually glancing at a subtraction chart right now. And there's one that I'm loving that pops out. That going diagonally down, on the triangular chart that I'm looking at, there's very clearly a 5 minus 3 is 2, 6 minus 4 is 2, 7 minus 5 is 2. Like, it's screaming, a little bit of constant difference to me. And how cool would that be, early, early, early, to just wonder like, "Oh, my gosh. Why are those always 2? And the kids... I think, kids could say,"Well, you're subtracting 1 more from 1 bigger of a number. That was poorly said, but.

Pam:That's interesting. Yeah, yeah.

Kim:Anyway.

Pam:Yeah, yeah.

Kim:(unclear) patterns that come out.

Pam:And those of you that are trying to teach constant difference later on, could pull up those. You know like, "Hey, look at all this diagonal. All the answers are 5. So, are you saying that 12 minus 7 is equivalent to 13 minus 8, is equivalent to 14 minus 9, is equivalent to 15 minus 10? Huh." And so, yeah, we're kind of nudging this idea that we can think of equivalent subtraction problems. That's nice.

Kim:Yep.

Pam:Yeah, super cool.

Kim:But, of course, our very favorite thing to do to work with basic facts or any mathematics are Problem Strings, right? (unclear). They build relationships. And so, we thought we would do a little bit with some Problem Strings today. I'm going to give you numbers this time, Pam.

Pam:Okay. Bring it on.

Kim:I'm always in the hot seat.

Pam:Alright, I got a pen in my hand.

Kim:Okay. Alright get a piece of paper.

Pam:This one's kind of a gnarly pen. It actually leaks a little bit. It's not my favorite.

Kim:Which is exactly why you shouldn't use pens.

Pam:But it's nice and bold. Alright, carry on.

Kim:Write harder. Okay, here we go. First problem.

Pam:[Pam laughs] Write harder. I like it. Okay, first problem.

Kim:Alright, 13 minus 3.

Pam:Okay, so if I was going to do this Problem String with kids. Because let's be clear, I don't really have to think about these problems very much. Okay. Then, I would expect students, if I'm doing this Problem String, that I would probably have a range of development in the classroom, and I might have some students going, "13, 12, 11, 10." Like, they might be counting back by ones. And we're expecting some 10's. I might have some kids thinking about the fact that it's 10 and 3. Or, you know, the meaning of the teen. If I sort of get rid of the 3 and 13 in 3 teen, then I'm left with the 10 leftover. But as a teacher, I would kind of establish that, and then I would. Kim, am I on a number rack or am I on a number line? Let's do both. If I was on a number rack, I'm going to assume that if I'm doing this Problem String, kids have done work on a number rack, so this is not their first shot at a number rack. So, I'm going to say, "How are you thinking about that?" And when a kid says, "Well, I removed 3 from 13," Then, I'm going to show 13 by 10 on the top and 3 on the bottom, and I'm going to say, "Oh, you just you got rid of that 3." And I'll move the 3 on the bottom back over to the other side. And I'm just left with 10 beads. If I was on a number line. So, it just depends on what my kids have had more experience with. If they had any experience with the number rack, I'll probably do it on a number rack. If they are ready to think about things on a number line, then I would say, "Oh, so you're saying 13..." And I would literally draw a number line, put a 13 on the right hand side. I would jump back one jump of 3, and I would land on 10.

Kim:Yep. Cool.

Pam:And that's it. Wouldn't spend a lot of time here. I would also, if I'm doing this problem, probably not have a kid talk a lot about the fact that they just removed 1 at a time. I would just, you know... Yeah. Okay, go ahead.

Kim:Yep. That's not going to be something you're modeling at this point?

Pam:No, not if I'm doing this whole string.

Kim:Yep. Okay. So, the next problem is 13 minus 5. What would you expect to hear and model for 13 minus 5?

Pam:Yeah. So, I might at this point, say, "I wonder if the first problem could help? Maybe? I don't know if..." You know, I almost don't finish my sentence a little bit. "I wonder if it could help you... Maybe? Go ahead. Use any any strategy you want." And then, I would let kids think and figure. I would watch to see who's counting by ones. I take note of that. And then, I might say, "Hey, did anybody use 13 minus 3 to help you with 13 minus 5?" And now, I'm looking for kids that. And then, I'll call on the kid who says. Now, it's still a little bit of a maybe. Maybe they did. Maybe they didn't. You're never quite sure with this age of kids. But I would try to pull out of a kid, somebody that said, "Well, if we know 13 minus 3 is 10..." And I would redraw that number line or I would re-put up the 10 beads and the 3 beads. And I would say, "So, we have 13. If I know 13 minus 3 is 10..." Move the 3 beads over, jump back the 3 on the number line, "...then I can think about 13 minus 5 as 2 more than that." So, if I'm on the number rack, I'd move 2 more beads on the top over, so I'm left with 8. I can clearly see those 5 beads and those 3 beads. Or I would remove 2, jump back a jump of 2 on the number line. And I would say,"Well, what is 2 back from 10? Oh, that's 8. And so that's how you guys are getting your 8. Does that make sense to everybody that we could subtract the 3 to get to 10, and then just subtract what's left over the extra 2?"

Kim:Yeah.

Pam:Cool.

Kim:Relating those two problems for sure.

Pam:Exactly, yeah.

Kim:What about 17 minus 7?

Pam:Again, I'm going to kind of watch. I'm not spend a lot of time here. Hopefully, kids are using the meaning of the teen. I think I might stick with the number line from here on out as I describe it, but it'll be similar. I'll start on 17. I'll have some kids suggest that I've subtracted 7 to get to 10, and land on the 10. If I may, my board this point, I've just... I'm actually drawing as I'm talking here. That number line that has the 13 is at a certain point... How do I say this. Left and right. And the 17 is about 4 to the right of that on the new number line. So, when I've drawn that new number line, I didn't start the 17 in the same place that I had started the 13. I've scootched it over to the right about 4. And my jump of 7, landed in the same location as the 10's were in the top two number lines. So, I've kept... Not vertically. Yeah, vertically. I've kept vertically aligned... Horizontally aligned? I don't know how to say that. The tick marks for the 10 are in the same horizontal position. Cut left to right, go in those. Now, I have three number lines, right, because I have the number line for the 13 minus 3, the 13 minus 5, and the number line for the 17 minus 7. I've landed on the 10 in the same spot. So, I've backed up 7 from that 17 that's further to the right. I know that was. It's hard to describe. But that's what it looks like. Okay.

Kim:Yep. Hey, I'm going to derail us for just a second.

Pam:Yeah.

Kim:Because I've heard people say. In fact, people have asked like, "When kids are representing their thinking, Do you require these specific directions about, like, how far and and does it need to perfectly line up?" Or teachers will say, "You know, I struggle, and I try." But I think maybe your point is that you're giving it a good go. It's not going to be perfectly proportional every single time, particularly if we're talking about kids, but the 17 should be the right of the 13, which should be... And the 10 should be to the left of the 13, and the 10s should line up together. (unclear).

Pam:(unclear). Absolutely. And that's only true for the teacher.

Kim:Yeah.

Pam:I think the student doesn't need to be thinking about that at all. The student is just using relationships. Now, if the student wants to draw a number line to jump to be able to keep track of their thinking, absolutely. But in that case, I'm not judging that kid's proportional jumps or where they land at all because they're using it to help them think. Now, I might look at what they did and help them find an error if they made a jump and, you know like, they subtracted 3 and they got to... I don't even know. And then, were like, "Hey, what do you know? What 14 minus 3 is?" Oh, yeah. That's 11, not 10." Or, you know, whatever. Like, that recording their thinking might be helpful. But no. So, yeah, good point. It's not about a perfect representation. It is about when the teacher is modeling student thinking, trying to get just as good as you can, so that the relationships pop a little bit more for students. If the relationships are up, they're lined up, the potential for the relationships to pop, so that kids can develop that number line in their head is a little bit stronger.

Kim:Cool.

Pam:Yeah.

Kim:Okay. Back on track. Sorry. We did 13 minus 3. We did 13 minus 5. We just did 17 minus 7. So, let's talk about 17 minus 8.

Pam:Cool. So, I'm going to, again, ask students, "Did anybody think about using 17 minus 7 to help you get 17 minus 8," and I'm going to look for a kid that does that. I'm going to redraw the number line for 17 minus 7 because the kid's going to say, "Yeah, because we knew 17 minus 7 was 10." I redraw that, and now it looks exactly like the one above it. And then, they're going to say, "But we need to subtract 1 more." So, I'll make a jump of 1 after that and land on 9. So, 17 minus 8 is 9. And now, I have four number lines on the board, and they've been sort of using the partner problem to help them with the second problem in the set.

Kim:Yeah. And I'm guessing, I'm betting that at some point, you're standing back, and you're like, "Hmm, that's interesting." And you're starting to maybe ask, or wonder, or put some words towards like, "I wonder what's going on here."

Pam:Yeah. Yeah, I might. Knowing my kids, I might ask somebody like, "Does anybody see a similarity about how so-and-so? Like Kim, used the 13 minus 3 to help get 13 to minus 5. And Aaron used 17 minus 7 to help him get 17 minus 8. Was there something similar about how they did that?" And I might try to pull some words out of kids. I might also say, "Huh, that's interesting that a lot of you got to the 10 first, and then just kind of subtracted the extra. Huh." I might just mention that, and then move on. Yeah,

Kim:Yeah, because in all four number lines, they'd all be lined up together, where there's a 10 on every single one of those number lines. I think that would really pop for your students. Alright, next problem. 15 minus 5.

Pam:Cool. I probably won't wait as long on this one. I'm watching kids. In fact, maybe I should say. If I saw a lot of kids counting by ones to get from the 13 minus 3 to 10, and the 17 minus 7 to 10, I might have also put... In fact, I might have put more emphasis on this generalization than the one you and I just talked about.

Kim:Right.

Pam:(unclear). I might have said, "Ooh, so like 13 minus 3, 10. 17 minus 7, 10. So, it's almost like we're thinking about the 10 and the teen, the extra 7, and we're just getting rid of it. So, it's like there's just this 10 showing up. That's what'teens' mean. Hmm. I wonder if anybody can use that to help you think about 15 minus 5, rather than... You know like, you can't count back, but can you just think about what 15 means to subtract 5?" Like, I might make that my emphasis. So, again, know your content, know your kids. And then, I will model starting at 15, and I'll jump back that 5. And I'll just note that my 10's are all lined up. So, the 15 is in between the 13 and the 17. Kind of. I'll be honest it's a little closer to the 13 than I wish it was. It's not right in the middle of a 13 and 17. It should be. It should be. But I'm writing in pen, so...

Kim:I was going to say too bad you don't have a pencil.

Pam:[Pam laughs] With kids, I'd be on a whiteboard, right? In fact, maybe that's worth mentioning. Teachers, if you're on a whiteboard, you can easily erase, and we recommend that. If you are on that document camera, don't be Pam and do it in a pen. Do it on a small dry erase board. Put a dry erase board underneath your... Or, be Kim and use a pencil because you do need to be able to erase. Though, don't you think, Kim, dry erase board is easier to erase than a pencil?

Kim:You know, honestly, it depends on the magnitude of the numbers because sometimes if I'm dealing with large numbers but also decimals like that's... You know, it gets a little bit more challenging. I want like a longer line to show a little bit more of the magnitude of the big numbers, so. And there's only so much space under a document camera with a dry erase. Dry erase markers are fat, and so...

Pam:I mean, you can get thin ones. You can get thin. I mean, you would want to. So, you're saying a pencil might be nice because you could draw finer stuff. Yeah. I also think it might depend on the reflection that you get off of the dry erase board under. Yeah. Alright, make that work. Okay, so we got 15 minus 5 is 10.

Kim:Okay. Last problem. 15 minus 8.

Pam:Cool. So, I'm going to ask, "Hey, did anybody do that thing that Kim did and Aaron did with this? Did anyboy use the 15 minus..." And I'm looking for glimmers of kids going, "Oh, yeah." And then, I might be like, "Oh, you were thinking about that? Even if you didn't finish, could you... Let's think that together. So, if 15 minus 5 is 10, but we need to subtract 8. How much more, so-and-so?" Whoever I'm working with. "Oh, yeah. So, we can just the extra 3? And what is what is 10 minus 3, everybody? We know that I Have, You Need. If I have 3, what do you need. Oh, 7. Sure enough." So 15 minus 8 is 7.

Kim:Mmhmm.

Pam:Yeah.

Kim:And so, at the end of the string, depending on is this your first go, is this the third time you've done this kind of string, will you talk just a minute about a little bit of a wrap up?

Pam:Yeah, absolutely. So, it kind of depends on what I want to emphasize with my students at this point. So, if a lot of them were counting by ones to do the 13 minus 3, the 17 minus 7, the 15 minus 5, I might make that be my point here. "Ya'll, like we have this pattern that if I have a teen, and I subtract off that bit of it that's the extra from the 10, what am I left? Wow, I'm always left with the 10. That seems important." Might that be something that I would pull out of kids, help them help me verbalize. But if that's not where most kids are struggling, then I might say, "Well, this is really interesting that we could get to the 10, and then just subtract the extra stuff. I wonder if that could help us with with problems in the future." So, in some way, we kind of verbalize, put in words, help kids sharpen their thinking by putting it in words the relationships that we're using in the string. In this case, can we get to the... If we're subtracting, can we subtract to the 10, and then just subtract what's leftover?

Kim:Yeah, which is one of our major relationships that we want kids to own for subtraction.

Pam:Absolutely. Yep.

Kim:So, it's a good experience.

Pam:Yeah.

Kim:I think we have time for one more experience.

Pam:Okay, cool, cool.

Kim:So, will you talk through these problems as well?

Pam:I'm ready.

Kim:Okay, new piece of paper and clunky pen. 12 minus 6.

Pam:And my pen just leaked. Your loving it right now. Okay, so 12 minus 6, if we're doing this Problem String, I'm not waiting very long on this. Doubles are things that kids kind of... Like, we want to do things to help our kids build doubles, but they are a thing that kids tend to know. So, I'll just usually, you know, "What is that?" Everybody's going to say 6. At this point, I'm going to draw a number line starting at 12. And I'm going to jump back of 6, and landing on 6.

Kim:Cool.

Pam:Okay.

Kim:Thirteen minus 6.

Pam:To be clear, though. Not because I said that. Because I'm asking a kid to say that, and then I'm representing their thinking, Yeah, okay.

Kim:Yeah.

Pam:So, sorry, what was the next one you said?

Kim:It's okay. 13 minus 6.

Pam:So, with 13 minus 6, I'm going to let them figure. Okay, that's the next problem. Let them figure. And I'm going to expect that there will be some kids counting back. I'm going to expect that there will be some kids kind of, you know like, nodding their head a little bit. And I can kind of almost see them counting back by ones. But I'm going to ask, "Did anybody use the problem before?" Like,"If you know 12 minus 6 is 6, can that help you with 1 more, 13 minus 6?" And I will probably even say that, "1 more, 13 minus 6." And then, as kids describe their thinking, I'm going to draw a number line with a 13 just to the right of that 12. About 1 to the right. And I'm going to say, "So, if I jump back that. If this jump is the same distance..." Try to make that really the same distance."...where would I land if I subtract the same but from 1 bigger? Where would I land?" And someone will say 7, and we'll land on that 7. So, I should have two number lines that look exactly the same, but one is shifted to the right 1. Where in the first one it's 6 to 12, and on the second one, it's 7 to 13. But the jumps are still all 6. Cool.

Kim:Which is why those attempt to be as proportional as possible is a huge goal because that gap or that minus 6 should look the same.

Pam:Yeah, absolutely. And teachers, if you're trying to do that, and you're like, "Oh!" You know like, in the in the bit of it, you're... It's not what you want. You can, at that moment, go "Whoa, this doesn't look right. Ya'll, help me. What should this look like?"

Kim:Yes!

Pam:And as the kids describe with the number line, you know like, "Oh, like the 13 should be to the right of the 12." "Well, then, guys, how big should the jump be? You know, when I drew it, I drew it way too big because... How should it relate to this jump of 6 up here?" I mean, as they discuss that with you, they're getting... It's not that they're really learning to model better, though they are, but they're really getting better at the relationships. They're owning like, "That should be the same length." Like, they're building spatial spatial relationships. Yeah, okay.

Kim:Yeah, I've seen you and I both do Problem Strings where we've sketched, you know, an area model or a number line, and then we look at it, and we've redrawn two or three times even. And the kids are involved in like, "Why is she fussing with this?" Like, What's wrong with this representation, so that she wants to redo it and make it better?" And we are totally talking about like, "Ooh, does that feel okay?" Like, "Does that seem close?"

Pam:Yeah. "Ooh, how would you change that?" And they're like,"That doesn't look like a rectangle at all." Like, it should be more square." Or,"That line isn't parallel." And you're like, "Oh." Like, "The sides of a rectangle should be parallel? Well, that seems important." Like, all that stuff can come in. Yeah. Okay.

Kim:Okay, cool. Ready for the next problem?

Pam:Yep.

Kim:16 minus 8.

Pam:Cool, another double. I'm not going to wait too long. I'm going to expect someone to say 16 minus 8. I'm going to draw a number line at 16 with a jump of 8 and landing on 8.

Kim:Yep.

Pam:Yep.

Kim:Cool. 16 minus 9.

Pam:Do you want to know that I just kind of... I drew the jump, and then realized that my jump was too short, so then I scratched out and made it a little bit longer. Anyway. Okay, so now 16 minus 9. I'm going to ask, "Hey, is anybody..." And I will actually suggest, as I say 16 minus 9, "I wonder if this problem could be helpful. I don't know. Maybe. Go ahead, and solve it any way you want." But then, I'm going to ask, "Did anybody use the 16 minus 8? And now, this time I'm going to start at the same 16.

Kim:Yeah.

Pam:And I'm going to say, "What should the jump look like? Should it be shorter or longer than the jump of 8? Oh, it's a jump of 9, it should be a little bit longer. How much longer? 1 longer. Well, if it's 1 longer, then where are we going to land? Ooh, we're going to land on 7. So, if I know 16 minus 8 is 8, then 16 minus 1 more than that should be 7." Cool. Again, pulling that out of kids, right? I'm trying to ask questions to get that for kids to say that, mmhmm.

Kim:Mmhmm. Next problem. 14 minus 7.

Pam:Again, a double. I'm not going to wait too long. Someone's going to say 7. I'm going to draw a number line with the 14, jump of 7, landing on 7.(unclear).

Kim:And 14 minus 5.

Pam:Fourteen minus 5. And in this case, again, I might say,"I wonder if anybody... Ooh, this seems a little different. It's not just 1 off, but that's okay. Go ahead." You know like,"Do whatever you want." Let them solve it. And then ask, "Did anybody use the 14 minus 7?" And try to draw out someone who says, "Well, this time I didn't subtract as much." "So, should the jump be shorter? longer? It should be shorter. How much shorter? It should be 2 shorter. Well, what is 2 to the right of 7? Oh, yeah. Sure enough, that's 9." So, if I've drawn that jump shorter, and I'm comparing it to the one above, I should be able to go, "Well, then, if it's 2 this way? What is 2 that way of 7? Sure enough, that's 9. Oh, nice." So, to finish up the string. "Wow, we really can use what we know. You guys, a lot of you are owning these doubles. Well, way to go. That's super that you know 12 minus 6. You know 16 minus 8. You know 14 minus 7. You know those doubles. Wow, knowing those, you can use what you know to just adjust either moving where you start or how big you jump. You can just adjust a little bit from when you know. Nicely done. Yeah, using doubles is super helpful." Cool.

Kim:Yeah, which is another one of our major strategies. And you have said a couple of times,"I'm not going to stay long on this particular problem because my kids know their doubles." And so, I can hear a teacher right now saying, "My kids don't know their doubles." And so, one of our favorite routines is similar to I Have, You Need. It's just a call and response, is we do a lot with Doubling and Halving because we know that kids have some intuitive intuition with Doubling. But we can strengthen that in moments of time, where we double numbers together, or we halve numbers together later on as they get older.

Pam:Absolutely.

Kim:Briefly, tell us what that looks like.

Pam:Well, and let me also mention that we can also work on a number rack is going to just scream. The doubles just scream. So, if I would have done that string on a number rack, as soon as I have 12 up there. Depending on how I put the 12, if I put it at 6 and 6, like were looking at the double of 6 being 12. And so, we want to... Work on a number rack can really help build doubles. But Doubling and Halving looks like literally... You know, I've got a little bit of time. I got 90 seconds before we're standing in line, the bell ring, whatever. And I might just say, "Hey, guys, random number. 15. What's double 15?" And then, I have to know my kids, so I have to know, do they have to think about that, but either way, once they tell it to me, then I ask them, "How are you thinking about that? How did you find that? Okay, you knew double 15. Double 14? Double 16?" So, I might string it a little bit. I might do numbers. And when I say"string it", you know like do a series of problems that are kind of related. So, if you know double 15, can it help you with double 16, could help you with double 17? So, yeah, you literally just throw out a number, ask them to double it, tell them, "How they're thinking about it?" And after kids have doubled a bit, a bunch over time, then a day I might go"Hey, we've been finding lots of doubles. Let's halve. Okay, guys ready? What's half of 24? What's half of 75?" Well, now, I would do some work out before I do 75. But they might say, "There is no half of 75. It's odd." And I might go, "If we have $75.00, can we split that? Oh!" And then, I want to sort of model the chunks they're doing. Anyway, so we can talk more about Doubling and Halving at some point, but that's a short routine that can help kids get facility with doubles?

Kim:Yeah. And once they own these strategies, right, once kids have had experience with them all these strategies that we've been mentioning over the last couple episodes, it's okay, in my mind, to say, "I see you counting, and I know you can count fast. You've been doing that a while, and so your fingers fly, but we have relationships that we can use, and so I want you to use them." You know?

Pam:Yeah. Hey, you reminded me. We were going to mention that in last week's episode about the Trinity story. Because literally, that happened, right? You had a group of first grade kids, and we were talking, and you said, "Yeah, Pam. These kids own the strategies. And they're just quick, so whenever they're... We clearly have developed strategies with them. But they count really quickly, and so, when we say, 'We're working with strategies,' they do them. But when we just give them a problem, or they're in the midst of another problem, they're deal is counting by ones." And I was like, "What are you going to do?" And you're like, "Oh, I'm just going to tell them to stop." And I was like, "What?" And so, you did, right? If I remember right, you pulled them together, and you said, "You guys, you own these strategies. When you see these problems, oh you're supposed to actually use the strategy?" And the kids were like, "Oh, we are?" And you're like, "Yeah."

Kim:Yeah.

Pam:Then, you did it with them. You're like, "Okay, so when when I see a problem like 13 minus 7, what's a strategy you could use?" And the kids start counting, and you're like, "No, no, no, no. We're not going to count on these anymore. I'm telling you, use these strategies." So, those of you that are pushing back and you're saying to yourself, "Pam, we don't want kids refiguring these problems all the time." There is a moment where you can say, "I don't want you refiguring it with a less sophisticated, therefore take too much cognitive load, strategy. I want you to actually dive into the strategy that might, in the moment, actually take a little bit more thinking because I want to build these relationships, then the facts become a natural outcome and so much more."

Kim:Yeah.

Pam:Yeah, nicely done. Alright, ya'll, let's help kids build their fact relationships and work with them enough to build, so that they have the facts at their fingertips. 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!

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