Math Facts Are Really Rectangle Relationships

Segment 9
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[00:00:00] Hey everybody. It is Cheri Dotterer here at Tier One Interventions podcast. It's great to have you here with us while we talk about the core of the math classroom. Today we're gonna be talking more about. Pizzas, rectangles, lockers. Oh my.

Cedric:  Back on March 10th, 2026, we started a 10 segment series Here is segment 9 from our October Workshop. If you are enjoying what you are learning, please subscribe and leave a comment.

I think that was the right order. Over to Jonily.

So when you're making rectangles and talking about which rectangles make squares as a level two expert, these are the numbers that you're going to do in your seasons. Or if you teach higher grade levels, all of these numbers might be in season one. Okay? So I teach algebra one and algebra two. I've already done all of these numbers with my [00:01:00] kids in season two.

I'm gonna pick a few new numbers, but I'm gonna still go back to all of these numbers that I've listed. So my kids are gonna get all of these numbers in all of the seasons.

So Amy says, okay, so what am I talking about here? When we pick a number? Like 24. Okay. Amy picks the number 24 with her first graders, and then she has them make all the rectangles possible. I listed some here. Okay. So Amy says, when she creates diamond problems, when she creates diamond problems, if I heard you right and correct me if I'm wrong, miss Amy, but when she creates a diamond problem for her first graders, she is going to be like, okay, eight and three, because her kids have been used to making rectangles with dimensions eight and three.

So eight and three, she's gonna have her kids make a rectangle that is an eight by [00:02:00] three. And then they're gonna have grid paper. So they're gonna see all of these, like I have eight columns. If I do all of these, this is actually dividing the bar. This is paper folding. And then I have three rows. So kids don't create this rectangle.

They actually on grid paper will show this eight by three using grid paper. We're doing lots of skills here and they're trying to figure out a, what eight plus three is, which is totally a first grade standard. Okay? And whatever they figure out for the sum of eight and three, that's gonna be this bottom number.

Okay? So we're practicing fluency and skill work through sensory reference tasks. And then whatever, eight and three, multiply. Now she's probably not saying multiply with her kids. Maybe she did. I don't know. It doesn't matter. But this top number is the multiplication of eight and three. [00:03:00] But this top number for first and second graders, I call the, you are finding the array number.

You are finding the array number. So first graders are still doing diamond problems. Thanks to Amy for teaching me this last time on how to deliver this to first and second graders, but this top number for first and second graders is the array number. I'm gonna show you that in a minute because this is another task these kids have done.

For the rest of us. I put D. D 'cause we're gonna, eight is dimension one, and three is dimension two on the dimension chart, meaning the length is one dimension and that is eight, and the height is the second dimension because I'm looking at a two dimensional shape. That's the three. This number is gonna be the array number because it's the number of blocks [00:04:00] or squares it took to make the rectangle.

Now, in third grade, this is going to become the area of the rectangle in third grade and beyond. This top number is the area of the rectangle, because area of rectangle is always product. It never changes. This is a transferable strategy, meaning the strategy doesn't expire when we get to new numbers or higher grade levels.

We want to be repetitively doing transferable strategies. So this number in the diamond problem becomes the multiplication, the product, all multiplication is area. Third grade is where the standard is for area. So prior to third grade, see, we're doing all of this, but this is the array number. Now how do I get an array number?

Amy mentioned the dimension [00:05:00] chart. Where does 24 go in the dimension chart? If I have my rectangle that has a length of eight, which we already saw vis visually, we build it with blocks, we drew it on grid paper. Now we're gonna do it a little more abstractly with a dimension chart. And Natalie, this is what the classroom teacher was talking about yesterday when we were doing the height and weight.

And she's oh my gosh, this is like the other thing you did with them. And IW and she couldn't remember what it was. And I said, it's the dimension chart. We have a three by eight. Now this is what's tough for younger kids, and this is why I did the height and the weight yesterday. They have to follow straight on the connection of both of these.

And then they have to circle the connector point. The connector box of eight and three, that connector blocks is block, is the array number of an eight by three rectangle. Now the array number. Is the product. [00:06:00] Later on it's gonna become the area. So if kids wanna tell me the array number is 11, I'm gonna say the array number is not 11, because eight plus three is 11.

That's only telling us how tall and how long to together. I'm gonna add eight and three. The array number is how many blocks did it take to make this rectangle? And then the kids have to figure out the array number is 24. Now, for all of you, this is a blank multiplication chart, but I call it a dimension chart because that's exactly how the multiplication chart is made.

And that's exactly what a multiplication chart is. A multiplication chart are the products of all of the different rectangles with these dimensions. My high school students now see the de see the multiplication chart. They're like, why didn't somebody ever [00:07:00] tell me this? Because I would know my math facts, my single digit math multiplication facts so much better if I knew all of this connection.

And that is why we do dimension chart. That is why we do making rectangles. And that is why we do locker problem. Because if I go to box 24 in locker problem, I wanna know all the students that touched Locker 24 student one did student two, did student three, did student four did student five did not touch Locker 24.

I cannot make a rectangle with an array number of 24 using a dimension five to get, whoops. 24. Jonily, you were doing so good. Okay. Using number five as a [00:08:00] height. Unless I go into fraction or decimal side lengths, which is fine. And we're going to eventually do I don't wanna say I can't make a rectangle with.

Dimension five. I just can't make one to put in the dimension chart with whole number dimensions. I can make a rectangle with height five, but it's just not gonna give me a whole number side length. And when we're focused on making rectangles in dimension chart, we're focused on whole number side lengths.

So I can't do it with a whole number. I can't land on 24 with student five. If I want whole number outcomes, I can do it later. I can do it later because I can have a non whole number, side length with a height of five with an area of 24. I'm not gonna worry about that right now. I can [00:09:00] do it, I just can't do it with whole numbers then.

Oh, but student six, compare with student four because a four by six rectangle makes 24 Oh. But what pairs with student three? So if student three touches, locker 24, student eight also has to touch locker 24. There's another rectangle eight by three. That's the one we charted. Another rectangle is 12 by two and 24 by one.

So all of the students that touch those lockers. Or the whole number dimensions of the rectangles when 24 is an array number or the area. So 24 is the array number, but there's lots of places in the dimension chart based on the locker problem that 24 can go. Guess what? We can do a six by four. So this is also the connector block for six by four.

This is also a 24. So the locker problem can tell us which students touch that locker. [00:10:00] Once we know which students touch that locker. Those are the whole number dimensions for the making rectangles. And then once I know all the dimensions, I can plot all of my locker numbers on the dimension chart. There are more locker numbers than others with 24, because 24 is gonna show up lots of times on the dimension chart.

11 isn't gonna show up very many times on the dimension chart. So I could ask another question. Where does 11 go on the dimension chart? Where does 11 go on the dimension chart? And how many times will we see 11 on the dimension chart? And

I lost my train of thought. How many times do we see 11 on the dimension chart? [00:11:00] And how many rectangles with whole number dimensions can be made with a area or array. Number 11.

And so now I can facilitate with these task dependent questions. The only way kids can answer these questions is if they have engaged in these tasks, locker problem making, rectangles, dimension chart diamond problems. The only way kids can answer these questions task dependent questions is if they've experienced these tasks.

And now when I say to students, what are the factors of [00:12:00] 24 at any grade level? And kids are like, I don't know. Now I have two choices. As a teacher, I can either be like you need to figure out what the factors are. Or I can just tell the kids the factors. Now I have a third door. Cheri knows what I mean by third door, the third way.

Okay. There's always a third way we think as educators that it's either this way or that way, and that those are all my options. There's always a third way. In the mastery math method. The third way is to support students through reference tasks. So when my kids struggle, even my high school kids don't know how to get the factors of 24 or don't know what factors mean, I can now say what dimensions?

Jonily, I wanna close out Tier One Interventions podcast, and that is, if you are listening to this podcast and you are [00:13:00] thinking, how do I get involved, you wanna head over to your show notes and click on the link. We are offering you to come to a session like this where you get all two and a half hours at one shot for $47.

You can then join another session for another $47. Or you can buy the whole year for $497 plus you have to buy the. When if you're going to go buy the whole year, you're gonna have to buy level one as well. But the coaching and these conversations that we're having are $497. That sounds to me like a really good deal for to, to really think the way you're thinking about not just mathematics, but occupational therapist delivery, speech therapy, delivery, [00:14:00] special ed delivery, and how we're helping these kids rethink, reregulate, relearn and think about life as whole.