# Ep 83: Single Digit Addition Facts Pt 3

January 18, 2022 Pam Harris Episode 83
Math is Figure-Out-Able with Pam Harris
Ep 83: Single Digit Addition Facts Pt 3

We build addition facts through relationships and fluency through experience.  So what important patterns and relationships can help students master or reason with their single digit addition facts rather than rote memorize? How can we teach those patterns and what experiences can help students automatize and become more fluent with addition facts? In this episode Pam and Kim dig deeper into single addition facts and how we as teachers can can empower students to reason like mathematicians.
Talking Points:

• Doubles are important
• The Shoe Game
• The Doubles Game (Cathy Fosnot)
• Near doubles
• Ten-and or The teens

Check out the patterns we talked about here:

And don't miss registration for one our deep dive workshops! https://www.mathisfigureoutable.com/workshops

Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

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

Kim Montague:

This week, we're going to continue our series on building addition facts through relationship and fluency through experience.

Pam Harris:

Ooo I like how you said that.

Kim Montague:

Oh, ok. Today we're gonna dive right into some of the most important relationships that we want to build in students.

Pam Harris:

Sorry for interrupting. I just really liked how you said that. That was good. "Building addition facts through relationship and fluency through experience", bam! Yeah. I work with good people. Alright. I want to start today, I just heard you say 'fluency'.

Kim Montague:

Yeah.

Pam Harris:

This word sticks in my craw just a little bit, because there's like a lot of definitions floating around. In fact, Jenny Bay Williams and I were at a conference one time, and I had only barely met her. She probably had no idea who I was. And I was like, "Hey, I have a problem with the way the math community at large uses that word fluency. And you're in a position to help change that." And brilliantly, they just wrote this whole - she and John San Giovanni just wrote Figuring Out Fluency, did a really nice job. Can I take credit? No, I'm not gonna take credit for that. But I did definitely like chat with her about this idea that the word fluency some people think means that I am good at procedures. And I want to push up against that and say, no, no, no. Fluency means that you don't get bogged down in it. That you are fluent enough in it, that you're not off to the side trying to figure it out. And now you can't get back to where you were, that it sort of drags you out of the work you were doing in such a way that you can't then have sort of the conversation if you're talking about being fluent in a language. Or that you can't do the mathematics because you sort of got derailed over here because it was bogging you down. Do we want students who are fluent? Everyone nod. Of course we do. But it doesn't mean that they are just like, oh, and now I can do the procedure without thinking. No, no, no, no, that's not fluent. That's a robot. That's a computer. We don't need those. We need thinking and reasoning human beings that don't get bogged down in stuff that we need them fluent at, fluent with. I just ended lots of sentences with prepositions. And I don't care.

Kim Montague:

You know it's so funny that you are mentioning this right now kind of off the cuff because it makes me think of one of my sons. Luke, my older son was in a situation where he was taking an IQ test like a legit psychological evaluation like the real deal IQ test, right?

Pam Harris:

My fave. Not.

Kim Montague:

I've spoken about Luke before and how you know, he's a thinker and reasoner and just amazes me right.

Pam Harris:

I'll pipe in. He is an amazing mathematician. I've had some wonderful math conversations with him. Far above where I would think a student in his grade level would be.

Kim Montague:

Yeah. Well, and so what I found interesting, so when we got the result of that back, one of his lowest, I'm air quoting, areas was fact fluency. It was a timed situation. So I already talked about this in our previous part of our series about -

Pam Harris:

How much we love time? We love - yah not so much.

Kim Montague:

But for such a great reasoner and thinker and you know, well accoladed for all things math. It was one of his lowest areas because it was speed dominant. It was completely based on his fluency. They quoted it fluency. And I would argue that he is very fluent with his facts.

Pam Harris:

I would agree with you completely. Yeah. In fact, if anything, I can almost picture him going, "7 x 8? That's it. Okay. That's all you want...okay. 56." Like he would be like trying to find what's hard in it.

Kim Montague:

Yes, what's the point?

Pam Harris:

"What am I supposed to think about here? Oh, nothing? Well, okay. I guess I'll just tell you that it's 56. Random." I could totally see him doing that. Yeah, now you know how we feel about IQ testing. Alright, so in anything where speed becomes like this important - now is speed ever important? Let's be clear. Probably on the basketball court. It's a little important.

Kim Montague:

When you're driving down the road.

Pam Harris:

When you're driving down the road. In fact my kids. I love my kids. Not all of them, but when they were learning to drive, some of them would slow down when doing things where they shouldn't slow down. Like you can't slow down here. I'm trying to think of it. What's a good instance?

Kim Montague:

Getting on the highway.

Pam Harris:

Yes, yes. Excellent. Or changing lanes.

Kim Montague:

Yeah.

Pam Harris:

Like, don't slow down. Like thou shall not slow down, not right now. And so they're like, "But I'm being careful." I'm like, "You're gonna get killed. This is not a time." So there are instances in life where speed is important.

Kim Montague:

But slowing down to be careful is a brilliant thought when you're thinking about math.

Pam Harris:

Yes, yes. Yeah, absolutely. Yeah. All right.

Kim Montague:

So onto patterns, Pamela.

Pam Harris:

Oh, yeah. So we're talking about patterns with addition facts, and that if we can help students identify patterns, in addition facts, then that will help them automatize those facts easier, better, and be more fluent in those addition facts. Because it takes the onus off of so many to like, get down, oh, no, I can just think about those. I understand that relationship, I can just use that relationship. I'll just do that. And then all of a sudden, we're sort of free to just think and reason and we're not bogged down by speed pressure, or by this sort of rote memory pressure. All we have to do is kind of use the relationships. And that is mathematizing, just as an aside, and so we can we could do it with single digit facts. Cool. One of those patterns that we want to bring up today is doubles. Doubles are important. Yes?

Kim Montague:

Doubles are really cool.

Pam Harris:

Yeah.

Kim Montague:

So we were -

Pam Harris:

Oh I have a story! Can I tell a story? Sorry.

Kim Montague:

Yah.

Pam Harris:

When I was beginning - okay, I'm gonna give you a little bit of a view into my non numeracy. When I started this journey. I was teaching high school math. I was doing technology training. I was wandering around the country helping teachers understand how to use technology to teach high school math better. I was doing a workshop I'll never forget in San Antonio, Texas with a group of PreCalculus teachers. And we were dealing with rational expressions. So a thing divided by a thing that looks like a fraction, but it had variables in it. And one of the teachers said something about something 35. And it had to do with if we doubled the 35, it turned into 70. And then there was a nicer division that happened. And I was like, wait, what? And somebody said something about doubles. And I was like, oh, yeah, doubles are really important. But 35? Like it had never occurred to me. When I was dealing with doubles, I was thinking about what we're talking about today, which is single digit doubles. And it never occurred to me that 35 was an important double. So we're not going to really go into higher digit doubles today. But we want kids to learn single digit doubles, because doubles are important. And then later then they can learn higher digit doubles.

Kim Montague:

So when you say we want kids to have some experience with them, some of that is just noticing them, right? Discussing that they're actually a thing in our lives, we have a lot of doubles, that if we just notice. And one of the things that we really love is games with doubles.

Pam Harris:

Yeah.

Kim Montague:

One of our favorites is a Cathy Fosnot game called the Shoe Game. So the mat, the play board is just pairs of shoes, kids roll a die that says maybe five, and then they move their marker ahead, they have to think about it first, right? Like what's the double of five, and so they have to consider 10. And so they move forward that many spaces. I love the differentiation of this game, because there's a second die available. And if kids don't know the double of five, then they can pick up the second die and turn it to the five and then they can count if they need to. So they count the five and five, or start with five and count up to 10. So they get to the end of the game board. They turn around, they come back and the game ends with both students being back at the beginning of the game board. So that's one that we absolutely love. Cathy Fosnot we'll give her credit for that game.

Pam Harris:

Yep, that's fantastic. Brilliant, brilliant curriculum writer KCathy is. Yeah. And so we would love for students to have some experience with games. Kim tell us about the, what is it called? Doubles?

Kim Montague:

Oh, Doubles. Yeah, there's a Doubles game. So a game board. This one you can start with single digit facts. They go up to 10 plus 10, I believe. So Doubles has a game board, that's like, I don't know, there's maybe 15,

Pam Harris:

It kinda looks like a bingo board. Right?

Kim Montague:

Yes, a bingo board. And it has just a bunch of single digit numbers on there. The kids roll -

Pam Harris:

Single and double, yeah?

Kim Montague:

I guess it goes up to 10, maybe 12?

Pam Harris:

So I've seen game boards that go up higher. So you sort of choose the game board that goes as high I guess as you want to.

Kim Montague:

So what I love about this game is that you can play with this particular board and play a couple of different games. So the Doubles Game is simply you draw a card. Oh gosh, I'm not even remembering it right.

Pam Harris:

I think that's why you were thinking about the lower ones. So you roll a die and that's gonna only go up till 12. But you could draw a card that can go up as high as the doubles of the your highest card. Or you can spin a spinner and I think that's the version I'm thinking of.

Kim Montague:

Yes, I remember now.

Pam Harris:

You can spin a spinner and then you have other options of numbers that you can double.

Kim Montague:

So if you spin a six, then you double it and you put your marker on the 12. And that's the first version. The next part I really love and it's more interesting to me is that then you can play Doubles Plus One or Doubles Minus One. So you spin the six you double it, you say 12. And then you say, I'm going to plus one or minus one to cover either the 13, or the 11.

Pam Harris:

Depending on what's free, right? And you can kind of think about that. And with any of those versions, we love the fact that then after the game is sort of played, then you ask questions like, "Hey, what numbers did you cover? What numbers did you not cover? What numbers will you never cover?" And now we get to start talking. So I'm going to let everyone think about that for a second. If we're playing the version where I spin the spinner, I double it, and I put my marker on that double, what numbers will I never cover on that gameboard, Kim?

Kim Montague:

Sorry I was distracted.

Pam Harris:

That's hilarious. If I'm playing the first version of the game, and I spin the spinner, and I double it, and I go cover that number. So we're just covering doubles, what numbers will I never cover?

Kim Montague:

I will never cover odds.

Pam Harris:

Odd numbers, right? Because you can't double whole number and get an odd number. And then if I'm playing the plus or minus version, now all of a sudden, I'm not covering any evens, right, I'm only covering odds. And that's so cool. And so we'd like games like that, that kind of have multiple levels of things going on and can kind of keep some kids thinking that have had more experience and keep kids really learning that have had less experience. Brilliant, brilliant game. One of the ways I like to develop maybe if your students don't know the doubles, so we kind of said in the Shoe Game, there was the option that they could sort of match the die. And then that would help them if they don't know the double. But in the Doubles Game, if they don't know it, then maybe you could notice that as a teacher. And a way to kind of help develop doubles, is to look at them on a number rack or a rekenrek .So for example, if I said, "Well, here's seven, what's double seven?" So I move over seven beads on the top. And then I would move over seven beads on the bottom. And then I would encourage kids to look at, to notice, the pattern of the 10 in there. So in seven, there's a five. And so in that other seven, since I'm doubling it, there's also a five, and I could pull out that 10. What's leftover? Well, the two and the two. Oh, so I can think about that as 10 plus that leftover four, and that can help me sort of think about doubling seven. So doubling single digit facts can be nice on a number rack, because you can sort of have those 10s kind of pop and it can be a way to develop them.

Kim Montague:

Yep.

Pam Harris:

Yeah. Cool. And we don't want students to just know doubles. We also want them to know near doubles. Right?

Kim Montague:

Right.

Pam Harris:

So sometimes I hear teachers say, doubles plus or minus one. I'm okay... okay, maybe I'm not. I was just about to say I'm ok with that. I would prefer if you don't mind, if you have to say doubles plus or minus one quickly move to near doubles.

Kim Montague:

Right.

Pam Harris:

So maybe it's - or just start with near doubles. So it's about near doubles, because then that kind of extends, I can get into bigger numbers now. And I can think about not just plus or minus one from a double. But I can think about questions like 32 plus 36. And I can say to myself, "Well, bam, that's almost 35 plus 35. But I'm down three and up one. So I'm really just down two and so if I know double 35 then I can just go down two. Then I can sort of think about 70 minus two, and I can get that 68, from a problem that was only near doubles. And so I kind of want to keep it as you know, general enough. Anyway, so near doubles, near doubles are important to work with youngsters.

Kim Montague:

Yeah. And it's really - what you're just describing is some early work into what could be the over strategy with near doubles. What you were just describing was early work into give and take.

Pam Harris:

Yeah, absolutely. Yeah. And so doubles are going to be really important to go there. Cool. What's another nifty pattern?

Kim Montague:

Yeah. So another pattern that we love is the 10 plus, or the teens, the 10-and. Which would have already happened in younger grades, right? So really early, we're talking about numbers composed of 10-and or we call those the teens and there's a whole weird, like names for all of them. So students will already know things like 10 plus five or 10 plus 4. Ten-and. But with the commutative property, remember we mentioned that earlier, that means they also know five plus 10 and four plus 10. So making them aware of oh, you know, the teen numbers so then, you know, the turnaround fact. And we have to do work to help them make sense of that and know that they know. Like, you know this and here's how you know this. And if you remember last week, we talked about allowing kids to mess with the number rack and flipping it upside down to help them cement that commutative property.

Pam Harris:

Yeah. So for example, Hey, let's just notice that we know 10. I pull over 10 beads on the top. Plus five. I pull over five beads on the bottom. We know that that's 15. So then, now I've flipped the number rack upside down. So do you also know what five on the top plus 10 on the bottom? Oh, sure enough. And so now we can kind of make sense of some facts that might - but what we don't want students going, oh, what's four plus 10? 4, 5, 6, 7, 8, and then counting up 10, when they know 10 plus four. They know 10 + 4 let's bring the community property in and notice that. Makes sense of it. We are good to go. Cool. So another cool pattern. I'm gonna let you talk about this one, because I think this might be one of your favorite patterns, is to think about how we can add nine. How do you add nine, Kim?

Kim Montague:

Yeah, so we just mentioned the commutative property right to think about plus 10. Like, if you have five plus 10, then if you know, five plus 10, then you can also just back up. It's the over strategy. It's one of my favorite strategies. So if you know five plus 10 is 15. Then you know, five plus nine is just one less. So 14, and that is early over strategy.

Pam Harris:

Bam. So just to back that up just a little bit. So if I'm dealing with a problem, like hmm, six plus nine. Then I might say to myself, "Well, I know 10 plus six, therefore I know six plus 10. (That's where you're talking about the commutative property.) So I know six plus 10. Well, that's 16. But I just need six plus nine. So backup one." Yeah. And it's funny, often, I'll talk to people and I'll say something about plus nine, and they'll go, "Well, there, there's this funny thing." They don't even necessarily describe it all that well, because they've kind of just been doing it in their head.

Kim Montague:

Yes.

Pam Harris:

We can make that much more - what's the word I want? Get it out of their head, make it not just something that's like living inside but we make it visible, we pull it out so we can talk about it and put it on a number line. Well, if I know six plus 10, big jump is 16. Then six plus nine, not such as big a jump. How not such a big a jump? Just one? One smaller than that? Yeah, sure. We just back up one. So totally. And you love the over strategy.

Kim Montague:

I do love the over strategy. Yes.

Pam Harris:

Such an important relationship. And so the relationships we talked about today so important to identify, explore, play with and own. Y'all in our next episode, we're going to talk about what's probably the most important relationship that we need kids to own as they are developing and really gaining and automatizing the single digit facts so you are not going to want to miss next week's episode as we talk about the partners of 10. Hey, we've got a free download for you today. We are loving really working with you all on the single digit facts, we have some important addition fact relationships, a free download for you, so that you can really like focus on these important fact relationships for addition. You can get that at mathisfigureoutable.com/addrelationships.

Kim Montague:

We also want to remind you that registration is currently open for our deep dive workshops, you can check those four workshops out at mathisFigureOutAble.com/workshops.

Pam Harris:

And if you're listening to this at a different time, you can totally join the waitlist at that same URL mathisFigureOutAble.com/workshops, that'd be a great great way to get on the waitlist or take the workshop now while registration is open. It's an experience that I don't think you can duplicate anywhere as we dive deep into content at your level. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!