This Paper Game Teaches Quadratics—Without Students Knowing!

June Event session 1 segment 2
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Cedric: [00:00:00] Welcome to Tier 1 Interventions odcast were we teach you inteventions techniques to keep your students in the classroom. We share adaptive resources that help students who struggle how they can also participate and gain the concepts being taught. This segment is from our live event held in June. Join me know as Jonily shares a math intervention that can be uses in every setting.

One of those things is are papers that look like this. So you're gonna need those, okay? You don't need any of the papers on the clipboard,

so nope. You're only gonna need the papers that look like this. You are also gonna need, there's two bags of red and white counters, and then while we're in the table basket, [00:01:00] there is a bookmark for each of you for your book. So everybody gets a bookmark. You get a bookmark, and you get a bookmark.

Perfect. And that gave you just a little bit of an idea of what we're gonna be using today, and I will very explicitly tell you what pieces and parts you need at each stage. But those wooden peg boards are game boards that look like this. So there are actually three game boards on the piece of paper.

You can either play individually or with someone. If two of you play, you can each be a color, one will be red and one will be white. If three of you play together, then that means two of you play and one [00:02:00] observes. If four of you play together, two plays, one observes, and one documents, I'll say all that.

Again, don't worry about it, and you don't have to do it exactly as I say. Okay? So these are three of the game boards. We're gonna model how the game is gonna be played. Then we're gonna put you into random groups that Audrey will explain. And once you're in those random groups, then I'm gonna give you about 12 minutes just to play and figure out an experience.

That's it. No other goal, right? So here's how it works. This is game board one, where you have one of each color. Red is on one side, white is on the other. The only goal is to trade places. Get the red where the whites are, get the white, where the red is. There are two types of moves. There's a jump and a slide.

Right now I can't do a jump because I, there's nothing here to jump over. The only move I can make right now is to slide either red or slide the white. So a slide is an adjacent spot, A spot directly [00:03:00] next to the color. The goal is in the fewest number of moves to get all of the colors to trade places.

The middle square starts empty and ends empty. So once you do that with this game board, then you go to the second game board and do the same thing. And things get more complicated as we go. For our primary kids, this is an essential game because they don't have access to board games. They should, they don't, and.

What holds them back from understanding number is counting one-to-one and using a board game to count spaces. So if kids are struggling, this is the experience you want them to have multiple times a year to get that one-to-one correspondence. As I said before, the math content goes all the way up to quadratic function, and you're gonna extract some of that as you play the game.

So in this case I can do a [00:04:00] jump. I could jump this red over here. Or I could slide. I could slide. So you could only use slides or jumps, and you can only jump over one piece. You can't jump over two at a time. So there's some parameters and rules here. Again, the goal is to trade those color places in the fewest number of moves.

If you start to slide or jump backwards, it's not the fewest number of moves start over. So again, that's why I said maybe one person plays, somebody watches somebody documents, or two people play. Somebody watches somebody documents. So in the end, you wanna get a rhythm playing the game. But you also want to document so that we can collect data for number of jumps, number of slides, and ultimately number of moves with one color, two colors, or three colors.

You can also do this with four colors, five colors, six colors, but we'll go as far as you want to today. Any questions about how the [00:05:00] game works?

Are you sure? Because I am not always the best explainer.

No. Very good question. Colors don't flip. So locker problem colors. Flip this, colors. Don't flip the red. Stay red, the white stay, right? So just pretend they're not two color counters. That's just what we're using. Great question. Other clarification? Yes,

you are allowed to move a piece multiple times. Just a little note, if you start moving the same piece too many times or backwards, you probably don't have the least number of moves. Other clarification?

Fantastic. Audrey. Come on up. I should stay here. Audrey is the absolute [00:06:00] queen of the kingdom with random grouping. So we've talked about building thinking classrooms and Peter, and there are lots of math resources. Audrey is the visionary and creator for random grouping cards. She's going to group us and she's gonna explain that.

Now, once she does that, we're gonna release you and you're just gonna play. I was gonna model it up here, but I think you guys are really intelligent because what I actually have up here are red and white papers and I, with students, I have them come up and model like in the front of the room. We're not gonna do that.

We're, I'm just gonna release you and I'm just gonna let the chaos ensue. So talk to us about your random group. Okay. Hi. I know Jon Lee's mentioned my name a couple times, but my name's Audrey Denberg. I am a math coach at Southwestern City School District. I've got a couple of my math teachers in the back.

Shout out to you guys. So I know John Lee talked a little bit about the purpose of random grouping cards or visibly random grouping cards. I've [00:07:00] put a little bit of a twist on them, so I know she's dropped Peter's name a lot. Do you guys know who she's talking about? Yes. I'll give you a hundred dollars if you can say his name, his last name correctly?

No. I don't know. Is it? I don't know. Okay.

And said it. So well, like I said, we put a little twist on this. If you've read Peter's book, building Thinking Classrooms, he does a lot of his grouping with playing cards. And if you do visible random grouping frequently enough, the kids will start to catch on that. There's only so many ways that you can group.

So in order to combat that so that students don't trade cards, we ended up, and I say we because I have a partner in crime her name's Rachel Sch Mullins. She also helps create these as well in our district. And these are a fancier, cuter way of grouping cards. So I'm gonna ask you guys to look at the front.

So this is what I'm categorizing as the front. You guys will notice that it does say a conference on it. So these were actually developed Rachel Sch Mullins and I ended [00:08:00] up presenting at NCSM, the National Council of Supervisors in Mathematics this past fall. And we are very themed in everything we do.

I think my whole back table back there can contribute to that. And so we had a field guide theme, hence this is where these kind of grouping cards came from. So there is nothing special about the front other than the fact that I love that I put a little goat on the mountain up top. I don't know if you guys noticed that.

Okay. Oh, all right. On the back is where all the magic happens. So can you guys actually flip over to the back of your card? Now, when I do this with students, I also do this with adults too. There's a lot happening. So whenever I introduce a new set of grouping cards, I always give time for us to decide how I could potentially group you.

So I'm gonna give you guys about 45 seconds at your table to try to figure out all the different ways I could group you go.[00:09:00]

And then I always try to instill facilitation moves in everything that I do. So I'm also gonna throw in a facilitation move. I'm gonna ask you to go ahead and select a spokesperson at your table so that when I randomly point at your table and say, tell me one thing, you're not all just sitting there awkwardly looking at me.

So go ahead. You've got 10 seconds. Decide who your spokesperson is.

Here we go. So again, please just give me one way that you could group. Okay. So let's go. Ladies in the back table, give me one way I could potentially group you today. In the back? Yep. All the way in the back. Okay. That's okay. Give me one way I could group you today the numbers. Okay. Let's go group in the front.

Gimme one way I could group [00:10:00] you type of flower. All right, let's go group in the front

type of vehicle. So there are some different camping vehicles. Let's go table in the middle. Okay. Different animal tracks. Let's go group up front.

So that's camping gear. Okay. I called them like their survival tools. Okay. Group in the back. Background color. There actually are some more ways, so I'll do rapid fire. You're allowed to just scream at me at this point. Go the snack. Okay. Our trail food. What else? Okay, I have two. You want me to come back to the circle thing in a second?

Anything else? Huh? Oh, he mentioned that up here? Yeah. Any other ways I could group you? Okay. Type of animals. Anything else? I think we actually got 'em all. Okay. So couple of things, and I am gonna go back to the circle. Okay. There are a couple [00:11:00] different ways that you could do this, but how I've done this on your grouping card is certain objects will group you based on threes.

Certain objects will group you based on fours. I'm the only one that knows that. Okay. So then I constantly switch it up for the kids. So that's how they can't figure out how I'm grouping them on what day. Okay. Now I'm gonna go back to the circle. Some of you it says winner. Some of you it says, I don't know, trail food.

I think some of you said, sorry, you're not a winner. So when I presented at NCSM one of the things that we were trying to bring as instructional coaches is how we bring what we defined as zest, meaning how we gamify and make things really fun and interactive for our teachers. So actually, if I was giving this to you guys to keep, this was actually a souvenir for every participant.

And so we put a sticker on, so it was a scratch off. So after at one point you scratched it off and then that's what told you, if you want something, so unfortunately you can't keep these. I'm sorry. So that's why it doesn't have a sticker on it. Okay. But that is actually not a way to group John Lee, I'm so sorry.

This is taking longer than I thought. Is that okay? Okay. Okay. Okay. Okay. So with that, now you're a little bit more comfortable with the grouping [00:12:00] card. So now I'm actually gonna tell you how to group. Okay. Now again, if you follow Peter, he also says that you need to make sure that you have some sort of visual cue for students to be able to go to their card.

Even as adults, I've seen this happen with administrators. Okay. I will shout out. Our administrators, I had shark themed grouping cards and I've never seen like a group of I don't even know what their ages are, like screaming at each other do you have this severed appendage? So that was one way to group was a severed appendage.

But what we have up around the roof, you guys take a look. Okay. We have some visual cue cards. So this is a way that you can get your students to go directly to the cue card rather than screaming at each other who has blueberries? Okay? Now this please keep in mind if you're in a classroom, we obviously would be in a much smaller room and you wouldn't have 48 different cards out there.

So this will take a little bit longer to get to our boards. I understand that. But if you're doing it in a classroom, I think some of my teachers can attest it goes a lot quicker, especially if your students are more familiar with it too. Okay? So with all that being said, [00:13:00] I am now gonna group you and then you're gonna head to your vertical surface, okay?

And I'm gonna group you based on trail food. So go find your trail food.

[00:14:00] Introduce yourselves. Talk for a moment.

Introduce yourself. Have a little small talk first. Yep. Chat it up a moment.[00:15:00]

Where are the potato chips? Alright, thank you.

Awesome. And hear my voice in 4, 3, 2, 1. Excellent. So basically you guys can just get started. You're gonna play the game. I'm gonna give you about 12 minutes, so you're gonna have a good extended period of time. So if you wanna continue to chit chat, that's fine. If you want to sit on the floor, that's fine.

If you want to move to a table now that you have your group, that's fine. We're not gonna necessarily utilize the vertical surfaces right now. We are gonna do that later. So it's up to you how you wanna arrange and how you want to play the game. Floors, tables, hallways. We're gonna go about 12 minutes, and if I look at my [00:16:00] clock,

we're gonna go till 10 36. So if you wanna go to the hallway, go ahead. 10 36. Have at it.

Lots of things right now because we wanna debrief this and we want to talk about the purpose and reality of it. Although it's a fun experience. There is so much cognitive science wrapped into this experience. This experience is not a one and done. This isn't like we're gonna do this, one class period, or for 30 minutes one day and never do it again.

To increase memory and retention of content. We use contextual, conceptual, sensory, visual experiences. Now, I know what you're gonna, I know what you're gonna say. Yeah, but how do we have time to do [00:17:00] this? This matters, and we're gonna talk about that throughout the day, because I'm gonna continue to refer to this exercise, this matters.

And a lot of the stuff that we do doesn't matter. We're doing it and we're not getting the result, but we can't let it go. Let it go. Okay, so first things first. If you did not give your random grouping card to Audrey, you can walk over there and give it to her now. So make sure Audrey's got all the cards.

Second, if you let me just do this. I need to stop talking. Tell me about my favorite three words. Tell me about your experience. Tell me about your experience. IW Yes. All right, Natalie, tell us about your experience. So this [00:18:00] was only my second viewing or interaction with this, but I hadn't really done it.

And one thing that surprised me and I wrote down to remember when I do with kids is my brain was like. I have to do red and white. Red and white and I can't jump white over white or red over red. Like it just made rules in my head that made sense. 'cause if you're playing a two person game, like sometimes, so I was like, oh, I can do this move.

But in my brain I had to train that out of me. And I know I teach sixth grade, they would, a lot of 'em would probably do the same thing. So being very clear what directions or just listening for or watching for those things as they're playing. Very good. Other things that you noticed or other phenomenon that happened?

Usually there's a winner to a game. So before I really understood what the game was about, I'm like, okay, so who wins? [00:19:00] Beautiful. Yeah, that's a great point. Stephanie, same. Okay. What else did you notice here and then back there. I. If you counted the slides, the correlation of counters correlates to the slides.

So if you have two counters, your two slides, if you have four counters, it's four slides. Six counters. Six slides. So if you would have eight counters, it would be eight slides.

Oh, we're clapping applause.

I want you to notice what just happened there, because I'm gonna refer to it later. So remember that experience, Sam. From the f facilitation side I listened and heard all the rules, like two player game, third person observe, fourth person write it down. And me and my group, we didn't do any of that. We're all math brain [00:20:00] people and all wanted our own board and we're like comparing.

But I just thought that was interesting because even as one of those good students who is paying attention, we did not play it like a game. And I appreciate that and I always make every single experience in my classroom with students that flexible. So remember the preschool coloring page where we're too explicit and that's the result we get, we box kids in.

So I'm very flexible to let them. Figure things out. If it's a rule that cannot be broken, absolutely, positively, first of all I wouldn't have any of that. I just wouldn't. But if it's a rule that absolutely positively, like you have to do this with one other person or what, I would make that very clear and explicit.

But I was pretty nonchalant about it. And I did that on purpose. I did that on purpose. Other things that you noticed? What [00:21:00] else can you tell me about this game? From my group, even after we finished one level, I wanted to see if I could get it in less moves. So before moving on, I did it like three or four times to see if I could do better.

Yes. We realized that anytime two of the same color were next to each other, then you were in trouble. Like it wasn't gonna work. You couldn't do anything. So we're like, okay, we know it has to be every other, but we still failed and we couldn't figure it out. It was driving us nuts like, but we knew they couldn't be next to each other.

Is that true or not true? Oh,

So there's a rule and there's a least amount of moves. [00:22:00] Other things you can tell me about this experience? Yeah, we started with the small one and then moved to each level and we started look to see if there the shortest moves, if there was some type of pattern. We noticed it was like 3, 5, 7, and then we were trying to see if we're looking for odd numbers, we were just looking for that kind of pattern.

So let me ask this, and this is an appropriate question for first grade, it's an appropriate question for third grade, it's an appropriate question for Algebra one high school. See, we oftentimes restrict what we do with kids because it's not it words on a standards page. We can teach the standards later.

As a matter of fact, our model and tier one interventions teaches us how to teach the standards at every grade level in 60 days or less. So that leaves us a lot of other days to do the right thing. All right, [00:23:00] so my point being we restrict ourselves and we restrict the kids when we say that doesn't specifically meet my standards, but it actually does because in the first grade standards in Ohio, the word iteration appears.

You may not have known that. So there are parts of the standards that we are either a misinterpreting. Or B, not even reading verbatim because it's not a letter and a number situation. It might be in the overall categories, it might be in the outline page of the standards. So when we think we're teaching the standards, we're actually not teaching the standards.

I wanna make that very clear. So I want you to think about this game, and I will tell you it's appropriate for every single grade level. But here's the [00:24:00] question that is appropriate for first grade, third grade, seventh grade, algebra, and high school. And I'm gonna ask it to all of you through this experience.

Is the phenomenon a linear, consistent experience? Or a non-linear experience. And I say that because we just talked about the patterns. And I'll say this to my second graders, was this a linear phenomenon or a non-linear phenomenon? And then I'll remind them, linear just means like the same increase of moves is happening.

The pattern is going up the same number of moves every time. And I'll just say that like that. So I can start to use the term linear a lot with every kid and every level. Non-linear means it's not just like fiction, nonfiction. We try to explain that and we, they have to have interactions over time with it.

So non-linear, just meaning the moves are gonna [00:25:00] go up by a different number each time. It's not gonna be consistent. So is this, and I gave this away to all of you because you're adults and I told you there's a quadratic phenomenon. I wouldn't have said that to kids. So you knew if quadratic that it's a non-linear phenomenon.

Also being quadratic, you may know that the number of moves forward and backward creates symmetry. There's always a middle type of move. What would that point be on a quadratic graph? Boom. It would be the vertex of the parabola function. Now just going to Cracker Barrel and playing this game, you wouldn't be like, oh, this is symmetrical and quadratic, and it would be the vertex of the you on my coordinate plane.

What I'm trying to share with you today is we are not going to extract all of the [00:26:00] mathematics. I want you to trust me that all of the mathematics is there, but because today is your first interaction with these experiences, we're only doing level one interaction with it. Natalie over there said, this was my second interaction .

Cedric: We will cut there for this week. Go be Awesome! Go Be Brilliant! We will see you in two weeks. Go enjoy some fantastic beach or mountain time. Jonily is in her happy place, Disney.