In a previous video, I talked about why I had such a problem with number bonds, and there was a good discussion around number bonds and it prompted me to want to talk to you about an alternative to number bonds.

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)

• The Problem with Number Bonds video: https://www.therecoveringtraditionalist.com/number-bonds-for-elementary-students/
• How To Use A Number Path: https://www.therecoveringtraditionalist.com/how-to-use-a-number-path/

Reminder About Why Number Bonds Suck

As you can see from my image here,

This is what number bonds look like, and really it can have any number in that spot there, but this example is with fractions, and the idea is that 3/4 is the whole and then it can be broken into two parts, or you might even have more than that. You could have three parts, four parts, it doesn’t matter.

But the problem I have here is that kids really struggle with how to put numbers in here and it doesn’t help kids out with knowing which circle is for the whole. Just because the lines connect them, our young kids especially don’t understand it.

One of the suggestions was to make that whole circle bigger. But one of the issues still is that the other two circles, the parts, are still the same size. So to me, when I look at that it makes me want to put the same amount into both of those circles, because they are the same size.

Models That Help Children See Magnitude Are Way Better

One of my big issues with number bonds is they don’t promote proportionality. It doesn’t help kids understand the magnitude of those parts. The model that I would like to encourage you to use as an alternative does.

Some people call these bar models, strip diagrams, part-part-whole models, they have lots of different terms, but the idea is that you use a rectangle to represent the whole, and then underneath or on top of you put two rectangles that would create that whole. Or you could have more rectangles, but we start usually with two parts and work our way there.

For these examples that I shared in the previous number bonds video, it’s just taking that idea and putting them into a rectangular format. This does a lot of things for our kids.

Number one, it helps build proportionality. Just by looking at that, I can tell that one of the parts that makes the seven needs to be a little bit bigger than the other part. And then I can think about and talk with my students about, what do you think these parts could be?

I want to show you two videos. If you want to watch this part, start the video above at 3:46.

I recorded my kids at home, because I have permission to record them so they’re always my video guinea pigs. And I was able to get my son, who is in kindergarten, he is a very early kindergarten kid, he’s born in August, he is only five years old and just turned five a month before kindergarten actually started.

Now, let’s just say he is NOT a normal kindergartener around mathematical thinking, because he does come from my household, I do know that. But I want to show you how he thought about this model. He has never used this model before, he’s a kindergartener, so keep all of that in mind. Here is what he did when I gave him a little worksheet that had these models. I really encourage you to watch it in the video above [WATCH AT 3:40-3:55], because it’s so cool to hear his thinking and see him. But if you can’t watch the video, here is what was said:

[Riggins] That’s a eight.

[Christina] Uh huh.

[Riggins] And then two.

[Christina] Ooh. ‘Cause eight and two makes?

[Riggins]10.

[Christina] 10? Okay. What would you do on this one?

[Riggins writes 1 and 9 in the parts]

[Christina] Ah. I’ve got a question for you. Why’d you put a one here and the nine there? It makes 10, you’re right, but how come you put the one there?

[Riggins] ‘Cause this space is littler.

[Christina] That space is littler, yeah? Okay, what are you gonna do on this one then? Hmm. There you go.

[Riggins writes a 10 and 0]

[Christina] Oh my goodness, how come you did that?

[Riggins] ‘Cause…

[Christina] ‘Cause why?

[Riggins] ‘Cause 10 plus zero makes 10.

[Christina] Yeah. But why’d you put zero here?

[Riggins] ‘Cause it’s a littler spot.

[Christina] It’s a little spot. That is great thinking, buddy.

Now again, remember, he is in the beginning of kindergarten, a young kindergartener, who has never used the model before, and he instinctively knew what to put in the model.

Now, understand, granted, a lot of kindergarteners don’t know that one and nine makes 10, all of that kind of stuff.

I knew his mathematical capabilities, so I picked 10. I knew that he knew a lot of his number combinations that make 10. I wasn’t trying to test him on the addition pieces, I just wanted to see what he would do with the model, because he’s never been exposed to that model. And even he knew that the smaller parts in the model means that it should be a smaller number, the bigger chunk should be a larger quantity.

That’s the power of this model that we don’t get in a number bond. That’s one of the reasons why number bonds are so darn confusing for our kids. Yet these two models still incorporate the same mathematics and build the same thinking for our kids but the part-part-whole model of this bar model is way more beneficial.

I do understand that on the first one he did not notice that they were the same size and he put an eight and a two. But I’ve gotta remember, he’s only five. And then with the other ones, he did notice the size difference and that helped him determine the quantities to put in there.

Okay, now, I took this to one of my older kids. If you want to watch this part, start the video above at 6:56.

Two of my kids don’t like being on video so they’re hard to get on video, so I got these two kiddos. The other one you’re going to see is my daughter. She is a fourth grader. Again, this is taken in the fall of her fourth grade year. And again, I knew that she knew the math, the addition piece of this, I’m not testing on her combinations to 100. I really am just looking at, does she understand the model without ever having been exposed to this model. Here’s her thinking.

[Christina] Have you used these models before?

[Sierra] I don’t think so.

[Christina] What do you think this is showing? Like, what’s your thought, just looking at it?[Sierra] Groups? Like you can break a hundred down into equal groups, so you’re like 50 and 50, and this one’s 70 and 30,

[Christina] Okay.

[Sierra] 60 and 40.

[Christina] Go ahead and fill it out, let’s see what you think.

[Sierra] Okay.

[Christina] Why’d you break that one into 50 and 50?

[Sierra] Because it looks split right here, like even. – Okay. And what do you think you’d do, so then this one you said?

[Sierra] 70 and 30.

[Christina] How come 70 and 30?

[Sierra] Because this one looks a lot smaller than this.

[Christina] Okay. And same thing there?

[Sierra] Yeah.

[Christina] You’d put a smaller amount, but not, so why is that one 40 this time?

[Sierra] Because this one is smaller than this one and this one is a little bit smaller than this one.

[Christina] Awesome, yeah.

And again, do you see it? It’s instinctual. I just asked her, what do you think this model is showing, and she was able to understand the model without any instruction on my end. That’s the power of a model, is that you want kids’ own thinking to be able to come through. What they’re thinking about should not be, “What does this model even do? What do I do here?” We want them to be thinking about the mathematics that they’re putting into that model. That is super, super huge, and I’ve gotta say that this model, the part-part-whole, bar model, whatever way you want to call it, does that so much more than a number bond does.

And that’s the big thing, is that even our youngest kids, when you see that visual, it’s telling you something about the size and magnitude of the numbers. And that to me is an awesome, awesome model, that even a kindergartener can instinctively understand it.

Not all of your kindergarteners are going to get it, I know that my kindergartener kiddo is a little bit different. You’re going to use smaller numbers, all of that, to start off with. But even just putting 3 in there as the whole and then having one part that’s smaller and one that’s a little bit bigger and helping them understand that it’s going to be a 1 and a 2 in there.

And if you notice, I know some people are going to ask this, I didn’t say anything to him about how that last model really couldn’t be zero. Technically he’s really wrong in that model, because if it was zero, the 10 would be all the way out and there would be no space for a zero. Right, does that make sense? But in his mind, he knew it was small. Well, he had already used the other small amounts. In the first one he used eight and two, then he used the one in the middle one, and then he’s like, okay, I know this one needs to be small as well, and the other small amount in his mind was zero. And to me that was huge. I wasn’t going to correct him and be like, no, buddy, you can’t do that. This is the first time he’s been exposed to this model, I want him to just get a feel of the model and have him use his own understanding for right now.

Now one word of caution, the model is still really abstract.

I mentioned this in the video about number bonds, but it’s really abstract because the kids don’t see the actual objects. They’re not seeing 10 things in there. You want to start out with making it very concrete, and then you work to representing it and then you move to the abstract. Again, you want to do all of these in the same activities.

Models That Connect To Future Models Are Way Better

A lot of times you’ll be having them model with objects, have them draw some kind of visual and then attach the symbols to it.

So, what that might look like is let’s say they’ve got their counting cubes out and they’ve modeled this, and again, we want to line it up, because guess what, guys, the other powerful part of this model is that it connects to another really, really powerful model which is the number line or a number path for young kiddos.

If you’re not familiar with a number path, I’ll also link to a video I’ve done about number paths and why we should be using number paths with our kindergarten and most of our first grade kiddos instead of number lines. In these examples I do show number lines, but just know, with kiddos that are young we are using number paths instead of the number lines.

They’ve modeled this, they’ve shown two and four, and then we’re drawing that, I’m doing a visual representation of that and I might even attach the whole, so do a visual of the two and four but also a visual of the six.

We want them to see the parts that come together to make the whole. It’s not just two and four, it is two and four, but two and four make six. We have got to reference both of those pieces, the parts and the whole. And then the cool part again is that we can draw a number line right underneath that, or if you’re using a number path have them model the same thing on a number path and have them see the two and the four that come together to make the six.

The power of the bar model and these part-part-whole models is that they’re linear and that linear leads into number paths and number lines.

Even as we work forward with our kids with fractions, you can still be modeling it concretely, and then we’ll draw, just trace right around it, attach those symbols to that drawing. Then again, you can attach another model to it. Help them make the connection between that model and other models that they are using. It’s super, super powerful.

The coolest part is that you can use these models for all kinds of mathematics. Here is a visual where they’re concretely modeling multiplication. They’re showing three groups of three concretely. We can attach the bar model to it, and then we can attach the number line to show the whole. Three, three and three becomes three, six, nine, as the whole. The coolest part is that they’re getting the proportionality, they’re connecting it to other really powerful models, and they’re able to use their intuitive sense with this that they can’t with a lot of other models.

Here’s the thing I don’t recommend, do not go out on the internet and go search for part-part-whole models, because most of them that you will find are not proportional. And the power of this model is that we want to change the parts so that they are proportional, that’s the big, big thing. A lot of the downloads that you’re going to find are not proportional, which makes the model not intuitive. Typically, the worksheets just have the parts split down the middle of the whole and that takes away the proportionality.

The cool part, I just made these models by making shapes, I’m on a Mac so I’m using Pages. If you have a Word document, find where your shapes are, create a rectangle. I drew one rectangle, and then I drew the smaller ones down here, and then the cool part is, now I can take that and I can drag it to make it a little bit different size. You can just drag and make them different sizes based upon what you want the focus to be on.

[Watch me doing it on my computer at 12:53 ]

Once you have your models made then you can change out the numbers. If I want to change this, and now we’re working with 24 here. And then do different ones. You can make them as big as you want. You can have this be a fraction, I can have this be three fourths. And then what are the parts here? You can put in the part here if you would like to do that. I could make this be one half, and it’s not showing up because my font may be a little bit odd.

But you guys get the picture. All I’m doing is just making rectangles and putting text inside of those rectangles, and then you can change the size of those rectangles based upon whatever you want to be having them focus on. See, they’re just rectangles. I’ve got two rectangles there and then I’m changing the size, lining them up so that they always end at the same spot there, but then you can adjust the size of them to be what you want.

So, that’s how you make your own. Don’t rely upon the worksheets out there that are part-part-whole, because most of them are incorrect representations, and we want to build that proportionality. That’s super, super huge.

Alright, I did title this An Alternative to Number Bonds, but you guys, I think that this model is THE alternative to number bonds. And I know that many of you might be tied to a curriculum that uses number bonds, and I just want to encourage you to replace number bonds with part-part-whole bar models, because you’re doing the exact same mathematics.

It is not about the model. They do not have to put it into a number bond to perform on an assessment. It’s not going to ask a kid on a state test, double check your state test, but every state test I’ve seen does not say, put this into a number bond. It wants kids to be able to decompose numbers and be able to compose numbers, put numbers together to create a whole. That’s the big idea. And this model can be the alternative that you use to number bonds.

I hope that this video has helped you build your math mind so that you can go build the math minds of your students. Have a great day.

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